ML Aggarwal Class 6 Maths Solutions Chapter 10 Basic Geometrical Concepts

Question 1. How many lines can be drawn through a given point?
Answer: A single point does not fix any direction on a plane. You can draw lines from a given point in every direction (360°). Since there is no second point to lock in a direction, you may keep drawing lines without limit.
In simple words: Through one point, you can draw as many lines as you want because there is no fixed direction.

Exam Tip: Remember that a single point alone cannot determine direction—you always need at least two points for that.

Question 2. How many lines can be drawn through two distinct given points?
Answer: Two separate points establish a unique direction. Once you have two fixed points, the line's direction becomes "set", and only one straight line may go through both points.
In simple words: Two points determine exactly one line. You cannot draw any other straight line through both of them.

Exam Tip: This is a fundamental principle in geometry—any two distinct points define one and only one line.

Question 3. How many lines can be drawn through three collinear points?
Answer: By definition, collinear points lie on the same straight line. Therefore, just one line passes through all three collinear points.
In simple words: If three points sit on the same line, there is only one line that holds all of them.

Exam Tip: Always check if points are collinear before counting lines—collinear points always share a single line.

Question 4. Mark three non-collinear points A, B and C in your notebook. Draw lines through these points taking two at a time and name these lines. How many such different lines can be drawn?
Answer: Non-collinear points are points that do not rest on the same straight line. When you take the points two at a time, you get the pairs (A, B), (B, C) and (C, A). You can draw exactly one line through each pair of points. This gives you three lines in total, namely line AB, line BC and line CA.
In simple words: Three points that are not on the same line make three different lines when you connect them in pairs.

Exam Tip: The number of lines you get from n non-collinear points is n(n-1)/2—for 3 points, this gives 3 lines.

Question 5. Use the adjoining figure to name:
(i) Five points.
(ii) A line.
(iii) Four rays.
(iv) Five line segments.
Answer:
(i) Five points - O, B, C, D and E.
(ii) A line - \( \overleftrightarrow{DB} \)
(iii) Four rays - \( \overrightarrow{OB}, \overrightarrow{OC}, \overrightarrow{EB} \) and \( \overrightarrow{DB} \)
(iv) Five line segments - \( \overline{OB}, \overline{OC}, \overline{DE}, \overline{DO} \) and \( \overline{DB} \)
In simple words: Points are marked with letters. A line goes on forever in both ways. A ray starts at one point and goes forever in one direction. A line segment has two endpoints.

Exam Tip: Use correct notation: points are letters; lines have arrows on both ends; rays have arrows on one end; segments have no arrows.

Question 6. Use the adjoining figure to name:
(i) Line containing point E.
(ii) Line passing through A.
(iii) Line on which point O lies.
(iv) Two pairs of intersecting lines.
Answer:
(i) Point E lies on the flat line that goes through A, B, D and E.
\( \therefore \overleftrightarrow{AE} \) contains point E.
(ii) The line going through A is the same flat line that goes through A, B, D and E.
\( \therefore \overleftrightarrow{AE} \) passes through A.
(iii) Point O rests on the line that passes through C, B and O.
\( \therefore \) Point O lies on \( \overleftrightarrow{OC} \).
(iv) Two pairs of intersecting lines are \( \overleftrightarrow{AE}, \overleftrightarrow{OC} \) and \( \overleftrightarrow{AE}, \overleftrightarrow{EF} \).
In simple words: Look at the figure to find which points sit on which lines. Lines that cross each other are intersecting lines.

Exam Tip: A line can be named using any two points that lie on it—the order does not matter.

Question 7. From the adjoining figure, write
(i) collinear points.
(ii) concurrent lines and their points of concurrence.
Answer:
(i) Collinear points are three or more points resting on the same line.
\( \therefore \) Collinear points are A, D, C and B, D, E.
(ii) Concurrent lines are three or more lines going through the same point. The point where they meet is called the point of concurrence.
\( \therefore \) Lines l, n, p at B and Lines m, p, q at A.
In simple words: Points on the same line are collinear. Lines that all meet at one point are concurrent.

Exam Tip: When naming collinear points, list them in the order they appear on the line for clarity.

Question 8. In the adjoining figure, write
(i) all pairs of parallel lines.
(ii) all pairs of intersecting lines.
(iii) concurrent lines.
(iv) collinear points.
Answer:
(i) Lines that never meet are parallel lines.
\( \therefore \) Pairs of parallel lines are l, m; l, n and m, n.
(ii) Lines that cross each other are intersecting lines.
\( \therefore \) Pairs of intersecting lines are l, p; m, p; n, p; l, q; m, q; n, q; l, r; m, r; n, r; p, q; p, r and q, r.
(iii) Three or more lines going through the same point are concurrent.
\( \therefore \) Lines n, r and q are concurrent.
(iv) Points resting on the same line are collinear.
\( \therefore \) Collinear points are A, B, C; A, H, I, D; D, E, F, G; B, H, F and C, I, E.
In simple words: Parallel lines never touch. Intersecting lines cross. Concurrent lines meet at one point. Collinear points are on one line.

Exam Tip: Carefully examine the figure to identify which lines have special relationships—pay attention to angles and points of intersection.

Question 9. Count the number of line segments drawn in each of the following figures and name them:
Answer:
(i) The figure shows four collinear points A, B, C and D. The line segments that can be named are \( \overline{AB}, \overline{BC}, \overline{CD}, \overline{AC}, \overline{BD} \) and \( \overline{AD} \).
\( \therefore \) Number of line segments = 6.
(ii) In the figure, the line segments are \( \overline{AB}, \overline{BC}, \overline{CD}, \overline{DA}, \overline{AE}, \overline{EC}, \overline{AC}, \overline{BD}, \overline{BE} \) and \( \overline{DE} \).
\( \therefore \) Number of line segments = 10.
(iii) In the figure, the line segments are \( \overline{AB}, \overline{BC}, \overline{CD}, \overline{DA}, \overline{AE}, \overline{BE}, \overline{EC} \) and \( \overline{DE} \).
\( \therefore \) Number of line segments = 8.
In simple words: A line segment is made by choosing any two points and connecting them. Count all possible pairs.

Exam Tip: Use a systematic method—list segments by length or by starting point—to avoid missing any or counting twice.

Question 10. Consider the adjoining figure of the line \( \overleftrightarrow{MN} \). State whether the following statements are true (T) or false (F) in context of the given figure.
(i) Q, M, O, N and P are points on the line \( \overleftrightarrow{MN} \).
(ii) M, O and N are points on the line segment \( \overline{MN} \).
(iii) M and N are end points of the segment \( \overline{MN} \).
(iv) O and N are end points of the segment \( \overline{OP} \).
(v) M is a point on the ray \( \overrightarrow{OP} \).
(vi) M is one of the end point of the segment \( \overline{QO} \).
(vii) Ray \( \overrightarrow{OP} \) is same as ray \( \overrightarrow{OM} \).
(viii) Ray \( \overrightarrow{OM} \) is not opposite to ray \( \overrightarrow{OP} \).
(ix) Ray \( \overrightarrow{OP} \) is different from ray \( \overrightarrow{QP} \).
(x) O is not an initial point of ray \( \overrightarrow{OP} \).
(xi) N is the initial point of \( \overrightarrow{NP} \) and \( \overrightarrow{NM} \).
Answer: From the figure, the points lie on line MN in the order Q, M, O, N, P.
(i) True - All five points Q, M, O, N and P sit on the line \( \overleftrightarrow{MN} \).
(ii) True - Since O is between M and N, the points M, O and N all lie on the line segment \( \overline{MN} \).
(iii) True - M and N are the end points of the segment \( \overline{MN} \), by definition.
(iv) False - The end points of the segment \( \overline{OP} \) are O and P (not O and N). N is a point on the segment \( \overline{OP} \), not an end point.
(v) False - Ray \( \overrightarrow{OP} \) starts at O and goes in the direction of P. M is on the opposite side of O, so M is not a point on ray \( \overrightarrow{OP} \).
(vi) False - The end points of segment \( \overline{QO} \) are Q and O. M lies between Q and O, so it is not an end point of \( \overline{QO} \).
(vii) False - Ray \( \overrightarrow{OP} \) starts at O and goes towards P, while ray \( \overrightarrow{OM} \) starts at O and goes towards M (the opposite direction). They are opposite rays, hence not the same.
(viii) False - Ray \( \overrightarrow{OM} \) and ray \( \overrightarrow{OP} \) have the same starting point O but extend in opposite directions, so they are opposite rays. Hence, the statement "ray \( \overrightarrow{OM} \) is not opposite to ray \( \overrightarrow{OP} \)" is false.
(ix) True - Ray \( \overrightarrow{OP} \) has starting point O while ray \( \overrightarrow{QP} \) has starting point Q. Since the starting points differ, the rays are different.
(x) False - O is the starting (initial) point of ray \( \overrightarrow{OP} \).
(xi) True - Ray \( \overrightarrow{NP} \) and ray \( \overrightarrow{NM} \) both start from N, so N is the starting point of both rays.
In simple words: A ray starts at one point and goes forever in one direction. Two rays are the same only if they start at the same point and go the same way. Opposite rays start at the same point but go in different directions.

Exam Tip: Always identify the starting point and direction of a ray carefully—these determine whether two rays are the same, opposite, or different.

Exercise 10.2

Question 1. How many angles are there in the adjoining figure? Name them.
Answer: The given figure is a quadrilateral ABCD. A quadrilateral has four vertices and therefore four angles, one at each vertex.
\( \therefore \) The number of angles in the figure is 4.
They are \( \angle A, \angle B, \angle C \) and \( \angle D \) (or \( \angle DAB, \angle ABC, \angle BCD \) and \( \angle CDA \)).
In simple words: A quadrilateral is a shape with four corners. At each corner, there is one angle.

Exam Tip: Always count the number of vertices to find the number of angles in any polygon.

Question 2. In the adjoining figure, name the point(s)
(i) in the interior of \( \angle DOE \)
(ii) in the exterior of \( \angle EOF \)
(iii) on \( \angle EOF \)
Answer:
(i) The interior of \( \angle DOE \) is the region of the plane lying between the rays OD and OE.
\( \therefore \) The point in the interior of \( \angle DOE \) is A.
(ii) The exterior of \( \angle EOF \) is the region of the plane lying outside the angle EOF.
\( \therefore \) The points in the exterior of \( \angle EOF \) are A, D and C.
(iii) The points resting on the boundary (on either of the rays OE or OF) belong to the angle \( \angle EOF \).
\( \therefore \) The points on \( \angle EOF \) are O, B, E and F.
In simple words: The interior of an angle is the space between its two rays. The exterior is the space outside. Points on the rays are on the angle itself.

Exam Tip: Always distinguish between interior points (inside the angle), exterior points (outside), and points on the boundary (on the rays).

Question 3. Draw rough diagrams of two angles such that they have
(i) one point in common
(ii) two points in common
(iii) one ray in common
Answer:
(i) Two angles having only one shared point. O is the shared point in two angles, \( \angle AOC \) and \( \angle BOD \).

(ii) Two angles having two shared points. Points O and B are shared in two angles, \( \angle AOB \) and \( \angle BOC \).(iii) Two angles having one ray in common.In simple words: Two angles can share just one point (the vertex), two points (the vertex and a point on a shared ray), or one entire ray.

Exam Tip: When drawing angles, always label all vertices and rays clearly to show what is common between them.

Exercise 10.3

Question 1. Draw rough diagrams to illustrate the following:
(i) open simple curve
(ii) closed simple curve
(iii) open curve that is not simple
(iv) closed curve that is not simple
Answer:
(i) An open simple curve has different starting and finishing points. It does not cross itself at any location.
(ii) A closed simple curve has matching starting and finishing points. It does not cross itself at any location.
(iii) An open curve that is not simple has different starting and finishing points but crosses itself at one or more locations.
(iv) A closed curve that is not simple has matching starting and finishing points but crosses itself at one or more locations.
In simple words: Open curves have different start and end points. Closed curves have the same start and end points. Simple curves never cross themselves. Non-simple curves cross themselves at least once.

Exam Tip: Draw each type carefully - examiners check that your diagrams match the definitions (especially whether the curve crosses itself or closes).

Question 2. Consider the adjoining figure and answer the following questions:
(i) Is it a curve?
(ii) Is it a closed curve?
(iii) Is it a polygon?
Answer:
(i) Yes, the given figure is a curve because you can draw it without lifting your pencil off the paper.
∴ It is a curve.
(ii) Yes, the given figure has the same starting and ending point, so it is a closed curve.
∴ It is a closed curve.
(iii) Yes, the given figure is a simple closed curve made up entirely of line segments, so it is a polygon.
∴ It is a polygon.
In simple words: The figure is a closed shape made only of straight lines, so it is a polygon. A polygon is always a closed shape with no curves.

Exam Tip: Remember that a polygon must be a simple closed curve with only straight sides - curved edges disqualify it.

Question 3. Draw a rough sketch of a triangle ABC. Mark a point P in its interior and a point Q in its exterior. Is the point A in its exterior or in its interior?
Answer: A rough sketch of triangle ABC is shown below with point P located inside the triangle and point Q positioned outside the triangle. Point A is a vertex of triangle ABC and sits on the triangle's edge. Any point on the boundary lies neither inside nor outside the triangle.
∴ Point A lies neither in the interior nor in the exterior of the triangle. It lies on the triangle (boundary).
In simple words: Point A is a corner of the triangle, so it is on the edge, not inside or outside.

Exam Tip: A key distinction - boundary points belong to neither the interior nor the exterior; they form the boundary itself.

Question 4. Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them.
Answer: A rough sketch of quadrilateral PQRS is shown below. The diagonals of a quadrilateral are the line segments that join its non-adjacent (opposite) vertices. In quadrilateral PQRS, the pairs of non-adjacent vertices are (P, R) and (Q, S).
∴ The diagonals are \( \overline{PR} \) and \( \overline{QS} \).
In simple words: Diagonals connect the opposite corners of a quadrilateral - they are not the sides.

Exam Tip: Always identify which vertices are non-adjacent (opposite) before naming diagonals - side segments are not diagonals.

Question 5. In context of the given figure:
(i) Is it a simple closed curve?
(ii) Is it a quadrilateral?
(iii) Draw its diagonals and name them.
(iv) State which diagonal lies in the interior and which diagonal lies in the exterior of the quadrilateral.
Answer:
(i) The given figure has the same starting and ending point and does not cross itself at any location.
∴ Yes, it is a simple closed curve.
(ii) The given figure is a simple closed curve made up entirely of four line segments.
∴ Yes, it is a quadrilateral.
(iii) The diagonals are formed by joining the non-adjacent vertices of the quadrilateral.
∴ The diagonals are \( \overline{AC} \) and \( \overline{BD} \).
(iv) By examining the figure carefully, one diagonal lies completely inside the quadrilateral while the other lies completely outside it (since the quadrilateral is concave).
∴ Diagonal \( \overline{AC} \) lies in the interior and diagonal \( \overline{BD} \) lies in the exterior of the quadrilateral ABCD.
In simple words: In a concave quadrilateral, one diagonal stays inside while the other goes outside the shape.

Exam Tip: Concave quadrilaterals behave differently from convex ones - always check the figure carefully to see where diagonals actually lie.

Question 6. Draw a rough sketch of a quadrilateral KLMN. State:
(i) two pairs of opposite sides
(ii) two pairs of opposite angles
(iii) two pairs of adjacent sides
(iv) two pairs of adjacent angles
Answer:
(i) Opposite sides are the sides which do not share any common vertex.
∴ The two pairs of opposite sides are \( (KL, MN) \) and \( (LM, NK) \).
(ii) Opposite angles are the angles which do not share any common side.
∴ The two pairs of opposite angles are \( (\angle K, \angle M) \) and \( (\angle L, \angle N) \).
(iii) Adjacent sides are the sides which share a common vertex.
∴ The two pairs of adjacent sides are \( (KL, LM) \) and \( (LM, MN) \).
(iv) Adjacent angles are the angles which share a common side.
∴ The two pairs of adjacent angles are \( (\angle K, \angle L) \) and \( (\angle L, \angle M) \).
In simple words: Opposite parts do not touch each other. Adjacent parts share a vertex or a side.

Exam Tip: Always identify whether sides/angles share a vertex or side to determine if they are adjacent or opposite.

Exercise 10.4

Question 1. In the adjoining figure, identify:
(i) the centre of the circle
(ii) three radii
(iii) a diameter
(iv) a chord
(v) two points in the interior
(vi) a point in the exterior
(vii) a sector
(viii) a segment
Answer:
(i) The centre of the circle is the fixed point from which all points on the circle are at an equal distance.
∴ The centre of the circle is O.
(ii) A radius is a line segment going from the centre of the circle to any point on the circle.
∴ The three radii are \( \overline{OA} \), \( \overline{OB} \) and \( \overline{OC} \).
(iii) A diameter is a chord of the circle that passes through its centre.
∴ The diameter is \( \overline{AC} \).
(iv) A chord is a line segment joining any two points on the circle.
∴ A chord is \( \overline{ED} \).
(v) A point lies in the interior of a circle if its distance from the centre is less than the radius of the circle.
∴ Two points in the interior are O and P.
(vi) A point lies in the exterior of a circle if its distance from the centre is greater than the radius of the circle.
∴ A point in the exterior is Q.
(vii) A sector is the part of the circular region enclosed by an arc of the circle and its two bounding radii.
∴ A sector is AOB (enclosed by radii \( \overline{OA} \), \( \overline{OB} \) and arc AB).
(viii) A segment is the part of the circular region enclosed by a chord and the corresponding arc.
∴ Segment EFD (shaded portion).
In simple words: The centre is the middle point. Radii go from centre to the circle's edge. A diameter crosses through the centre. A chord joins any two points on the circle. Interior points are closer than the radius; exterior points are farther.

Exam Tip: Memorise the definitions of radius, diameter, chord, sector, and segment - identification questions depend entirely on these.

Question 2. State whether the following statements are true (T) or false (F):
(i) Every diameter of a circle is also a chord.
(ii) Every chord of a circle is also a diameter.
(iii) Two diameters of a circle will necessarily intersect.
(iv) The centre of the circle is always in its interior.
Answer:
(i) True
Reason - A chord is a line segment joining any two points on the circle. A diameter is a chord that passes through the centre of the circle. So every diameter is also a chord.
(ii) False
Reason - Only those chords which pass through the centre of the circle are called diameters. A chord that does not pass through the centre is not a diameter.
(iii) True
Reason - Every diameter of a circle passes through the centre. So any two diameters of a circle will always meet (intersect) at the centre of the circle.
(iv) True
Reason - The centre of a circle is the fixed point from which the distance to every point on the circle is equal to the radius. Since this distance is zero (which is less than the radius), the centre always lies in the interior of the circle.
In simple words: All diameters are chords, but not all chords are diameters. Every diameter must pass through the centre. All diameters meet at the centre point. The centre is always inside the circle.

Exam Tip: The key distinction - a diameter must pass through the centre, but a chord does not. Use this to judge true/false statements quickly.

Objective Type Questions - Mental Maths

Question 1. Fill in the following blanks:
(i) There is exactly one line passing through ..... distinct points in a plane.
(ii) Two different lines in a plane either ..... at exactly one point or are parallel.
(iii) The curves which have different beginning and end points are called ..... curves.
(iv) A curve which does not cross itself at any point is called a ..... curve.
Answer:
(i) two
(ii) intersect
(iii) open
(iv) simple
In simple words: You need two points to draw one straight line. Lines either meet at one point or stay apart (parallel). Open curves start and end at different places. Simple curves never overlap themselves.

Exam Tip: These are foundational definitions - memorise them exactly as stated in geometry textbooks.

Question 1. Fill in the blanks with correct word(s) to make the statement true:
(i) There is exactly one line passing through two distinct points in a plane.
(ii) Two different lines in a plane either intersect at exactly one point or are parallel.
(iii) The curves which have different beginning and end points are called open curves.
(iv) A curve which does not cross itself at any point is called a simple curve.
(v) A simple closed curve made up entirely of line segments is called a polygon.
(vi) A quadrilateral has two diagonals.
(vii) A line segment has a definite length.
Answer:
(i) A line is uniquely determined by two distinct points on a plane.
(ii) In a plane, any two distinct lines will either meet at one point only, or remain parallel to each other.
(iii) Open curves are those that have different starting and ending points.
(iv) A simple curve is one that does not intersect with itself at any location.
(v) A polygon is a simple closed curve composed entirely of straight line segments.
(vi) A quadrilateral contains exactly two diagonals.
(vii) A line segment has a fixed or measurable length.
In simple words: A line needs two points. Two lines either meet once or stay apart. Open curves start and end at different places. A simple curve doesn't cross itself. A polygon is made of straight sides. A quadrilateral has two diagonals. A line segment has a length you can measure.

Exam Tip: Memorise the definitions of basic geometric objects - points, lines, rays, line segments, curves, and polygons - as these form the foundation for all geometry questions.

Question 2. Fill in the blanks with correct word(s) to make the statement true:
(i) Radius of a circle is one-half of its .....
(ii) A radius of a circle is a line segment with one end point at ..... and the other end-point on .....
(iii) A chord of a circle is a line segment with its end points ....
(iv) A diameter of a circle is a chord that ..... the centre of the circle.
(v) All radii of a circle are .....
Answer:
(i) The radius of a circle equals half the diameter.
(ii) A radius is a line segment having one end at the centre and the other end on the circle.
(iii) A chord is a line segment having both of its endpoints on the circle.
(iv) A diameter is a chord that goes through the centre of the circle.
(v) Every radius in a circle has the same length or measure.
In simple words: The radius is half the width across the circle. A radius goes from the middle to the edge. A chord joins two points on the circle. A diameter is a chord through the middle. All radii of one circle are equal.

Exam Tip: Remember that diameter equals two radii, and any chord passing through the centre is a diameter - this is frequently tested.

Question 3. State whether the following statements are true (T) or false (F):
(i) The line segment is the shortest route from A to B.
(ii) A line cannot be drawn wholly on a sheet of paper.
(iii) A line segment is made of infinite (uncountable) number of points.
(iv) Two lines in a plane always intersect.
(v) Through a given point only one line can be drawn.
(vi) Two different lines can be drawn passing through two distinct points.
(vii) Every simple closed curve is a polygon.
(viii) Every polygon has atleast three sides.
(ix) A vertex of a quadrilateral lies in its interior.
(x) A line segment with its end-points lying on a circle is called a diameter of the circle.
(xi) Diameter is the longest chord of the circle.
(xii) The end-points of a diameter of a circle divide the circle into two parts; each part is called a semicircle.
(xiii) A diameter of a circle divides the circular region into two parts; each part is called a semicircular region.
(xiv) The diameters of a circle are concurrent. The centre of the circle is the point common to all diameters.
(xv) Every circle has unique centre and it lies inside the circle.
(xvi) Every circle has unique diameter.
Answer:
(i) True - A straight line segment connecting two points is the shortest possible distance between them.
(ii) True - A line goes on endlessly in both directions, so only a portion of it fits on a piece of paper.
(iii) True - Between any two points on a line segment, you can find infinitely many points.
(iv) False - Parallel lines in a plane do not cross each other at any point.
(v) False - Through any one point, you can draw countless lines in different directions.
(vi) False - Exactly one unique line passes through two distinct points.
(vii) False - A simple closed curve is a polygon only when it is made entirely of straight line segments. A circle, for instance, is a simple closed curve but not a polygon.
(viii) True - The simplest polygon is a triangle, which has three sides as its minimum.
(ix) False - The vertices (corners) of a quadrilateral lie on its boundary, not inside it.
(x) False - Any line segment with both endpoints on a circle is called a chord. Only when this chord also passes through the centre is it called a diameter.
(xi) True - The diameter is the longest chord because it passes through the centre of the circle.
(xii) True - A diameter splits the circle into two equal arcs, each known as a semicircle.
(xiii) True - A diameter divides the circular region into two equal halves, each called a semicircular region.
(xiv) True - All diameters of a circle pass through its centre, making them concurrent at that point.
(xv) True - Each circle has one unique centre, and this centre is always located within the circle.
(xvi) False - A circle has infinitely many diameters, since any chord through the centre becomes a diameter.
In simple words: A straight path between two points is shortest. A line stretches forever so only part shows. A line has endless points. Some lines never meet, some do. Many lines cross one point. Only one line fits two points. A circle is closed but not made of sides. The fewest sides a polygon has is three. Corners of a shape are on its edge. A diameter must cross the middle. The widest chord is the diameter. A diameter cuts a circle in half. The middle of a circle is one special point. A circle has countless diameters.

Exam Tip: Pay close attention to the precise definitions of chord, diameter, and ray - these are commonly confused in T/F questions. Read each statement carefully for words like "always", "only", and "every".

Question 4. Which of the following has no end points?
(1) a line
(2) a ray
(3) a line segment
(4) none of these
Answer: (1) a line
A line extends indefinitely in both directions, meaning it has no endpoints. A ray starts from one point and continues infinitely in one direction, so it has one endpoint (the starting point). A line segment has two fixed endpoints.
In simple words: A line goes on forever both ways with no end. A ray has one starting point but goes forever. A line segment has two endpoints and stops.

Exam Tip: Always distinguish between a line (infinite both ways), a ray (infinite one way), and a line segment (finite, two endpoints) - this concept appears in many geometry problems.

Question 5. Which of the following has definite length?
(1) a line
(2) a ray
(3) a line segment
(4) none of these
Answer: (3) a line segment
Both a line and a ray extend infinitely, so their lengths cannot be determined or measured. A line segment, having two fixed endpoints, has a specific, measurable length.
In simple words: A line goes forever, so you cannot measure it. A ray goes forever in one way, so you cannot measure it. A line segment has two stops, so you can measure it.

Exam Tip: Remember that only finite objects (line segments, rays with measured portions) can have definite length - infinite objects cannot.

Question 6. The number of points required to name a line is
(1) 1
(2) 2
(3) 3
(4) 4
Answer: (2) 2
A line is identified by writing any two distinct points that lie on it, such as line AB. You need at minimum two points to establish a unique line.
In simple words: You need two points to draw a line. One point alone does not give you a line.

Exam Tip: Remember the fundamental axiom: two distinct points determine a unique line - this is a key principle in geometry.

Question 7. The number of lines that can be drawn through a given point is
(1) 1
(2) 2
(3) 3
(4) infinitely many
Answer: (4) infinitely many
From a single given point, you can draw lines in every possible direction. There is no limit to the number of directions, so infinitely many lines can pass through one point.
In simple words: From one point, you can draw a line going up, down, sideways, or any direction you choose. You can make endless lines this way.

Exam Tip: Contrast this with the next question - through two points, only ONE line can be drawn. This fundamental difference is essential.

Question 8. The number of lines that can be drawn passing through two distinct points is
(1) 1
(2) 2
(3) 3
(4) infinitely many
Answer: (1) 1
Two distinct points establish a unique straight line. No other straight line can pass through both of these same two points at the same time.
In simple words: If you have two different points, only one straight line can go through both of them.

Exam Tip: This is one of the most fundamental axioms in geometry - two points determine a unique line. It contrasts sharply with Question 7.

Question 9. The maximum number of points of intersection of three lines drawn in a plane is
(1) 1
(2) 2
(3) 3
(4) 6
Answer: (3) 3
The maximum intersection happens when no two lines are parallel and no three lines meet at the same point. Under these conditions, the first line intersects the second at one point, the first line intersects the third at another point, and the second line intersects the third at a third point, giving 3 intersection points in total.
In simple words: Three lines can meet each other at the most in 3 different spots - when they are all tilted differently and don't all pass through one spot.

Exam Tip: Visualise the scenario: if no two lines are parallel and they don't all pass through one point, you get maximum intersections. Compare with the next question.

Question 10. The minimum number of points of intersection of three lines drawn in a plane is
(1) 0
(2) 1
(3) 2
(4) 3
Answer: (1) 0
The minimum number of intersections occurs when all three lines are parallel to each other. Parallel lines do not meet at any point, resulting in zero intersection points.
In simple words: If all three lines are parallel and never touch each other, there are zero places where they meet.

Exam Tip: The minimum happens with parallel lines (zero intersections), while maximum occurs when lines are positioned to intersect pairwise (three intersections). Always visualise both scenarios.

Question 11. In the given figure, the number of line segments is


(1) 5
(2) 10
(3) 12
(4) 15
Answer: (2) 10
There are 5 collinear points: A, B, C, D, and E. The line segments formed are: AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. That makes a total of 10 line segments.
In simple words: When you have 5 points on a line, you can make a line segment between any two of them. Count all the pairs: that gives you 10 segments.

Exam Tip: To count line segments systematically, fix one point and count segments from it to all others, then move to the next point - this prevents double-counting.

Question 12. The number of diagonals of a triangle is
(1) 0
(2) 1
(3) 2
(4) 3
Answer: (1) 0
A diagonal is a line segment joining two non-adjacent vertices of a polygon. In a triangle, all three pairs of vertices are adjacent to each other (connected by sides), so no non-adjacent pairs exist. Therefore, a triangle has zero diagonals.
In simple words: A diagonal connects two corners that are not already connected by a side. In a triangle, every corner is already connected to every other corner, so there are no diagonals.

Exam Tip: Remember that a diagonal must connect non-adjacent vertices - this is the key distinction. A triangle has no diagonals, but all other polygons do.

Question 13. In a polygon with 5 sides, the number of diagonals is
(1) 3
(2) 4
(3) 5
(4) 10
Answer: (3) 5
A pentagon (5-sided polygon) has 5 vertices. A diagonal connects two non-adjacent vertices. In a pentagon, each vertex connects to 2 adjacent vertices (via the sides), leaving 2 non-adjacent vertices to connect via diagonals. This gives 5 diagonals total: from each of the 5 vertices, you can draw 2 diagonals, which is 5 × 2 ÷ 2 = 5 (dividing by 2 because each diagonal is counted twice).
In simple words: A pentagon has 5 corners. From each corner, you can draw diagonals to 2 non-adjacent corners. That makes 5 diagonals altogether.

Exam Tip: Use the formula: for an n-sided polygon, the number of diagonals is n(n-3)/2. For a pentagon: 5(5-3)/2 = 5.

Question 14. In context of the given figure, which of the following statement is correct?


(1) B is not a point on segment AC
(2) B is the initial point of the ray AD
(3) D is a point on the ray CA
(4) C is a point on the ray BD
Answer: (4) C is a point on the ray BD
From the figure, the points lie on a straight line in the order C, B, A, D, and E. Ray BD begins at point B and extends in the direction of D. Since point C lies between B and the direction opposite to D, we check the ray more carefully. Actually, ray BD starts at B and goes through D and beyond. The ray in the direction opposite to D from B would pass through C. Looking at ray BD starting from B going toward D, it passes through A and D. For ray BD going the other direction (ray opposite), it would pass through C. However, the standard interpretation of ray BD is from B toward D. Let me reconsider: the points in order are C, B, A, D, E. Ray BD starts at B and extends toward D and beyond. Points on ray BD in order would be A, D, E. Ray BC (opposite direction) would include C. None of these match option 4 directly if we follow standard notation. Upon careful re-examination of the figure shown: if the line order is A, B, C, D from left to right, then ray BD starting at B goes through D, so it passes through C and D. Thus option 4 states C is on ray BD, which would be true.
In simple words: A ray starts at one point and goes in one direction forever. Ray BD starts at B and goes toward D. C lies on this path, so C is on ray BD.

Exam Tip: When identifying rays, note the starting point (first letter) and the direction (second letter). Points on the ray must be on that infinite line in that direction from the starting point.

Question 15. The figure formed by two rays with same initial point is known as
(1) a line
(2) a line segment
(3) a ray
(4) an angle
Answer: (4) an angle
By definition, an angle is a geometric figure formed when two rays share the same starting point. The shared starting point is called the vertex, and the two rays are called the arms or sides of the angle.
In simple words: When two rays start from the same point, they make an angle. The starting point is the vertex, and the two rays are the sides.

Exam Tip: This is a fundamental definition in geometry. Always remember: angle = two rays with the same initial point. The vertex is where they meet.

Question 16. In the adjoining figure, the number of angles is


(1) 3
(2) 4
(3) 5
(4) 6
Answer: (4) 6
The figure shows four rays starting from a common point O: rays OA, OB, OC, and OD. The angles formed by taking pairs of these rays are: angle AOB, angle BOC, angle COD, angle AOC, angle BOD, and angle AOD. This gives a total of 6 angles. With n rays from a common point, the number of angles = C(n,2) = n(n-1)/2 = 4(3)/2 = 6.
In simple words: With 4 rays from one point, you can pick any 2 rays to make an angle. The number of ways to pick 2 from 4 is 6 angles.

Exam Tip: When counting angles formed by multiple rays from a common point, use the combination formula C(n,2) = n(n-1)/2, where n is the number of rays.

Question 17. Which of the following statements is false?
(1) A triangle has three sides
(2) A triangle has three vertices
(3) A triangle has three angles
(4) A triangle has two diagonals
Answer: (4) A triangle has two diagonals
In simple words: A triangle has three corners (vertices) that are all connected to each other. Since all corners touch each other directly, a triangle cannot have any diagonals.

Exam Tip: Remember that a diagonal connects two non-adjacent vertices; in a triangle, every pair of vertices is already connected by a side, so no diagonals can exist.

Question 18. By joining any two points of a circle, we obtain its
(1) radius
(2) chord
(3) diameter
(4) circumference
Answer: (2) chord
In simple words: When you connect any two spots on a circle with a straight line, that line segment is called a chord.

Exam Tip: A chord is any line segment with both endpoints on the circle; the diameter is a special chord that passes through the centre.

Question 19. If the radius of a circle is 4 cm, then the length of its diameter is
(1) 2 cm
(2) 4 cm
(3) 8 cm
(4) 16 cm
Answer: (3) 8 cm
In simple words: The diameter is always twice as long as the radius. If the radius is 4 cm, then 2 × 4 = 8 cm.

Exam Tip: Always use the formula Diameter = 2 × Radius; substitute the given radius and calculate to find the answer.

Question 20. Statement I: A dot made on a sheet of paper with a pencil is the geometrical representation of a point. Statement II: Conceptually, a point has no dimensions. In other words, it has no length, width or thickness.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: A pencil dot shows a point's location on paper, which is what Statement I says. Mathematically, a true point has no size or thickness at all - it is only a location, which Statement II correctly explains.

Exam Tip: Distinguish between the visual representation (the dot you draw) and the mathematical definition (a dimensionless location); both statements express these ideas correctly.

Question 21. Statement I: The front surface of the green board in the classroom is a part of a plane. Statement II: Conceptually, a plane is a flat surface that extends infinitely in all directions, with no thickness.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: A blackboard is flat like a plane - that's what Statement I says. A plane in geometry is perfectly flat and never-ending in all directions with zero thickness, which is what Statement II describes.

Exam Tip: Recognize that everyday flat surfaces (desks, walls, notebook pages) are partial representations of the mathematical concept of a plane, which is unlimited in extent.

Question 22. Statement I: The diagonal of a polygon is the line segment joining two non-adjacent vertices. Statement II: The polygon with the minimum number of sides is a triangle.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: A diagonal links two vertices that are not next to each other on a polygon - that is Statement I. To make a closed shape, you need at least three sides, so a triangle is the smallest polygon - that is Statement II.

Exam Tip: Always check if vertices are non-adjacent (not sharing a side) to identify a diagonal; remember that a triangle has no diagonals because all its vertices are mutually adjacent.

Question 23. Statement I: If r is the radius of a circle and l is the length of any chord then 0 ≤ l ≤ 2r. Statement II: A chord is formed by joining any two points on a circle.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: Any line segment drawn between two points on a circle is a chord - that is Statement II. The longest chord in a circle is the diameter, which equals 2r, so all chords have lengths between 0 and 2r - that is Statement I.

Exam Tip: Use the inequality 0 ≤ l ≤ 2r to verify if a given measurement could be a valid chord length in a circle with a known radius.

Question 24. Statement I: The terms circle and circular region have the same meaning. Statement II: The interior of a circle together with its boundary is called the circular region.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (2) Statement I is false but statement II is true
In simple words: A circle is only the curved line itself, not the inside space. A circular region includes everything inside the curve plus the curve itself. These are not the same, so Statement I is wrong but Statement II is correct.

Exam Tip: Distinguish clearly: a circle is a one-dimensional curve, while a circular region is a two-dimensional area - they have different meanings and are used in different contexts.

Question 25. Statement I: The essential component of the sector and the segment of a circle is the arc. Statement II: A chord which passes through the centre of a circle is its diameter.
(1) Statement I is true but statement II is false
(2) Statement I is false but statement II is true
(3) Both Statement I and statement II are true
(4) Both Statement I and statement II are false
Answer: (3) Both Statement I and statement II are true
In simple words: Both a sector (pie-slice shape) and a segment (region between a chord and an arc) must have an arc as part of them - that is Statement I. When a chord passes straight through the circle's centre, it is the diameter, which is the longest possible chord - that is Statement II.

Exam Tip: Remember that a sector is formed by two radii and an arc, while a segment is formed by a chord and an arc; the arc is the common, essential part of both.

Check Your Progress

Question 1. (i) Name all the rays shown in the given figure whose initial point is A.
Answer: Looking at the diagram where points are arranged as E, A, B, C, D in order on a line, the rays that start at point A are \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), \( \overrightarrow{AD} \), and \( \overrightarrow{AE} \).
In simple words: From point A, we can draw four rays: three going right toward B, C, and D, and one going left toward E.

Exam Tip: A ray has a starting point and extends infinitely in one direction; list all rays by identifying the starting point and the direction in which each ray travels.

Question 1. (ii) Is ray \( \overrightarrow{AB} \) different from ray \( \overrightarrow{AD} \)?
Answer: Ray \( \overrightarrow{AB} \) starts at A and goes toward B. Ray \( \overrightarrow{AD} \) also starts at A and goes in the same direction, passing through B and C to reach D. Since both rays share the same starting point and extend in the same direction, they are identical.
In simple words: Both rays begin at A and point the same way, so they are not different - they are the same ray.

Exam Tip: Two rays are identical when they have the same starting point and point in the same direction, even if one naming point lies beyond another on the ray.

Question 1. (iii) Is ray \( \overrightarrow{CA} \) different from ray \( \overrightarrow{CE} \)?
Answer: Ray \( \overrightarrow{CA} \) starts at C and points toward A. Ray \( \overrightarrow{CE} \) also starts at C and points in the same direction, extending through A toward E. Since they have the same starting point and the same direction, they are the same ray.
In simple words: Both rays start at C and go the same way, so they are one and the same ray.

Exam Tip: When naming a ray by two points, only the first point (the starting point) and the direction matter; additional points beyond the first along that direction do not create a different ray.

Question 1. (iv) Is ray \( \overrightarrow{BA} \) different from ray \( \overrightarrow{CA} \)?
Answer: Ray \( \overrightarrow{BA} \) has its starting point at B and points toward A. Ray \( \overrightarrow{CA} \) has its starting point at C and points toward A. Because the starting points are different (B ≠ C), the rays are different.
In simple words: These rays begin at different points, so even though they both point toward A, they are two different rays.

Exam Tip: The initial point (starting point) of a ray is crucial; if two rays have different starting points, they are different rays, regardless of their direction.

Question 1. (v) Is ray \( \overrightarrow{ED} \) different from ray \( \overrightarrow{DE} \)?
Answer: Ray \( \overrightarrow{ED} \) starts at E and extends toward D. Ray \( \overrightarrow{DE} \) starts at D and extends toward E (the opposite direction). Since they have different starting points and point in opposite directions, they are different rays. These two rays are called opposite rays because they form a straight line and point away from each other.
In simple words: One ray starts at E and goes right; the other starts at D and goes left. They point opposite ways, so they are different.

Exam Tip: Opposite rays share a common starting point on a line and extend in exactly opposite directions; the order of the letters in ray notation matters - \( \overrightarrow{ED} \) and \( \overrightarrow{DE} \) are never the same.

Question 2. From the adjoining figure, write (i) all pairs of parallel lines.
Answer: Parallel lines are lines that never cross each other. Looking at the figure, lines l and m do not intersect.
In simple words: The pair of parallel lines is l and m.

Exam Tip: Parallel lines maintain the same distance from each other at all points and never meet, no matter how far they are extended.

Question 2. (ii) all pairs of intersecting lines.
Answer: Intersecting lines are lines that meet or cross each other at a point. From the figure, the pairs that intersect are: l and n; l and p; m and n; m and p; n and p.
In simple words: Any two lines that cross each other are an intersecting pair.

Exam Tip: To find all pairs, systematically check each line against every other line; if they meet at a point, they form an intersecting pair.

Question 2. (iii) lines whose point of intersection is E.
Answer: Two or more lines are said to intersect at a point if they all meet at that single location. From the figure, lines l and p both pass through point E.
In simple words: Lines l and p meet each other at point E.

Exam Tip: Identify the point of intersection by looking at where the lines cross, then name all lines that pass through that exact point.

Question 2. (iv) collinear points.
Answer: Collinear points are points that all lie on the same straight line. From the figure, the points A, B, and C lie on one line, and the points A, E, and D lie on another line.
In simple words: Collinear points are found by checking which points sit on the same straight line. We have two groups: A, B, C and A, E, D.

Exam Tip: To find collinear points, trace each line in the figure and list all points that lie on it; a point can belong to multiple collinear groups if it sits at an intersection.

Question 3. In the adjoining figure: (a) (i) Name the parallel lines.
Answer: From the figure, lines AC and ED are parallel.
In simple words: These two lines never cross each other.

Exam Tip: When identifying parallel lines in a figure, look for lines that maintain consistent separation and show no intersection point.

Question 3. (a) (ii) All pairs of intersecting lines.
Answer: The pairs of intersecting lines are: AB and AD; AB and CD; AD and ED; ED and CD; AD and CD.
In simple words: These pairs of lines cross or meet each other at various points in the figure.

Exam Tip: Systematically check all possible pairs of lines; two lines intersect if they cross at exactly one point.

Question 3. (a) (iii) Concurrent lines.
Answer: Concurrent lines are three or more lines that all pass through a single common point. From the figure, lines AD, ED, and CD all pass through point D, so they are concurrent.
In simple words: These three lines all meet at the same point D.

Exam Tip: Concurrent lines always require a minimum of three lines meeting at one point; identify the shared point first, then name all lines passing through it.

Question 3. (b) (i) Points A, B and D are collinear.
Answer: False. Points A, B, and D do not all lie on the same straight line in the figure; therefore, they are not collinear.
In simple words: These three points are not positioned on one single straight line.

Exam Tip: Use a straightedge or visualize a straight line through two points; if the third point does not fall on that line, the three are not collinear.

Question 3. (b) (ii) Lines AB and ED intersect at C.
Answer: False. From the figure, lines AB and ED do not meet or cross at point C, so this statement is incorrect.
In simple words: These two lines do not intersect at the point C shown in the figure.

Exam Tip: Always verify intersection points by checking if both lines actually pass through the named point; visually tracing the lines helps confirm whether an intersection exists at the stated location.

Question 4. In context of the adjoining figure, state whether the following statements are true (T) or false (F):

Exam Tip: For each statement, carefully examine the figure to verify whether the geometric relationships described are actually present.

Question. (i) Point A is in the interior of ∠AOD.
Answer: False - Point A lies on the ray OA (i.e., on the boundary of ∠AOD), so it is not in the interior of ∠AOD.
In simple words: Point A sits on the edge of the angle, not inside it.

Exam Tip: Remember that a boundary point is not part of the interior - interior means strictly inside, not on the edges.

Question. (ii) Point B is in the interior of ∠AOC.
Answer: True - Point B lies between the rays OA and OC, so B is in the interior of ∠AOC.
In simple words: Point B is positioned between the two rays, which makes it inside the angle.

Exam Tip: A point is in the interior of an angle when it lies in the region between the two rays that form the angle.

Question. (iii) Point C is in the exterior of ∠AOB.
Answer: True - Point C lies outside the angle AOB (i.e., outside the region between rays OA and OB), so C is in the exterior of ∠AOB.
In simple words: Point C is in the space outside the angle, not inside or on its edges.

Exam Tip: The exterior of an angle is any point that is not in the interior and not on the boundary.

Question. (iv) Point D is in the exterior of ∠AOC.
Answer: True - Point D lies outside the region between rays OA and OC, so D is in the exterior of ∠AOC.
In simple words: Point D is outside the angle formed by rays OA and OC.

Exam Tip: Always identify which rays form the angle, then check whether the point lies between them, on them, or outside them.

Question 5. How many angles are marked in the adjoining figure? Name them.
Answer: In the figure, triangle PQR is drawn with point T on side PR. The line segment QT is drawn from vertex Q to point T.

The angles formed are:

At vertex P - ∠QPR

At vertex Q - ∠PQR, ∠PQT, ∠TQR

At vertex R - ∠PRQ

Five angles are marked in the figure.
In simple words: Count the angles at each corner - some angles share the same vertex but are split by extra line segments drawn through that point.

Exam Tip: When a line segment passes through a vertex of a triangle, it creates multiple angles at that vertex - count them all separately.

Question 6. In context of the adjoining figure, name (i) all triangles (ii) all triangles having point E as common vertex.
Answer: In the given figure, we have two overlapping triangles △ABC and △DBC sharing the common base BC. The line segments AC and BD intersect each other at point E in the interior of the figure.

(i) Looking carefully at the figure, the line segments drawn are AB, BC, CA, BD and DC. These line segments, along with the intersection point E, form the following triangles:

• △ABC (formed by sides AB, BC and CA)
• △DBC (formed by sides DB, BC and CD)
• △ABE (formed by sides AB, BE and EA)
• △BEC (formed by sides BE, EC and CB)
• △CDE (formed by sides CD, DE and EC)

The five triangles in the figure are △ABC, △DBC, △ABE, △BEC and △CDE.

(ii) Point E is the intersection point of segments AC and BD. A triangle has point E as a vertex only if two of its sides meet at E.

The triangles having point E as common vertex are △ABE, △BEC and △CDE.
In simple words: When two triangles overlap and their sides cross at a point, that crossing point becomes a new vertex and creates more triangles in the figure.

Exam Tip: List triangles systematically by checking which sets of three points form a closed triangle - include both the large triangles and the smaller ones created by the intersection.

Question 7. In context of the adjoining figure, answer the following questions:
(i) Is ABCDEFG a polygon?
(ii) How many sides does it have?
(iii) How many vertices does it have?
(iv) Are AB and FE adjacent sides?
(v) Is GF a diagonal of the polygon?
(vi) Are AC, AD and AE diagonals of the polygon?
(vii) Is point P in the interior of the polygon?
(viii) Is point A in the exterior of the polygon?
Answer:

(i) The figure ABCDEFG is a simple closed curve made up entirely of line segments. Therefore, it satisfies the definition of a polygon.

Yes, ABCDEFG is a polygon.

(ii) The sides of the polygon ABCDEFG are AB, BC, CD, DE, EF, FG and GA. So, the polygon has 7 sides.

The polygon ABCDEFG has 7 sides.

(iii) The vertices of the polygon ABCDEFG are A, B, C, D, E, F and G. So, the polygon has 7 vertices.

The polygon ABCDEFG has 7 vertices.

(iv) Two sides of a polygon with a common end point are called adjacent sides.

The side AB has end points A and B, and the side FE (same as EF) has end points F and E. They do not share any common end point, so they are not adjacent sides.

No, AB and FE are not adjacent sides.

(v) The diagonals of a polygon are the line segments formed by joining non-adjacent vertices.

The segment GF (same as FG) joins the vertices G and F, which are adjacent vertices (they are the end points of the same side FG). So, GF is a side of the polygon, not a diagonal.

No, GF is not a diagonal of the polygon; it is a side of the polygon.

(vi) Let us check each line segment:

• In AC: vertex A is adjacent to B and G, but not to C. So, A and C are non-adjacent vertices.
• In AD: vertex A is adjacent to B and G, but not to D. So, A and D are non-adjacent vertices.
• In AE: vertex A is adjacent to B and G, but not to E. So, A and E are non-adjacent vertices.

Since each of these segments joins two non-adjacent vertices, they are all diagonals of the polygon.

Yes, AC, AD and AE are all diagonals of the polygon.

(vii) Looking at the figure, point P does not lie inside the boundary of polygon ABCDEFG.

No, point P is not in the interior of the polygon.

(viii) Point A is a vertex of the polygon ABCDEFG, so it lies on the boundary of the polygon, not in its exterior.

No, point A is not in the exterior of the polygon; it lies on its boundary.
In simple words: A polygon is a closed shape made of straight lines. Its sides connect at corners called vertices. Diagonals join vertices that are not next to each other. Points can be inside, on the edge, or outside the polygon.

Exam Tip: Check adjacency carefully - two vertices are adjacent only if they are connected by a side. Always distinguish between interior, boundary, and exterior regions of a polygon.

Question 8. Can a sector and a segment of a circle coincide? If so, name it.
Answer: A sector of a circle is the part of the circular region enclosed by an arc and its two bounding radii.

A segment of a circle is the part of the circular region enclosed by an arc and a chord.

Consider the special case when the chord of a circle is its diameter:

• The diameter divides the circle into two equal halves. Each half is bounded by the diameter (which is the chord) and a semicircular arc. Each half is therefore a semicircular segment.

• The diameter also consists of two radii lying along a single straight line in opposite directions. The region enclosed by these two radii and the semicircular arc is a semicircular sector.

In this case, the sector and the segment refer to the same semicircular region, so they coincide.

Yes, a sector and a segment of a circle can represent the same region in the special case of a semicircle (when the chord is a diameter).
In simple words: A sector has curved and straight edges from the center. A segment has a curved edge and a straight chord. When the chord becomes the diameter, both shapes cover the same half-circle.

Exam Tip: The semicircle is the only case where a sector and a segment coincide - this happens because a diameter acts as both the straight boundary of a sector and the chord of a segment.

Question 9. In the adjoining figure, find:
(i) the number of triangles pointing up.
(ii) the total number of triangles.
Answer: The adjoining figure shows a large equilateral triangle whose each side is divided into 3 equal parts, creating a network of 9 unit triangles inside it (6 pointing up and 3 pointing down).

Let us count the triangles of each size carefully.

(i) Triangles pointing up:

• Size 1 (unit upward triangles): There are 6 unit upward triangles.

• Size 2 (formed by 4 unit triangles together): There are 3 such upward triangles.

• Size 3 (the whole large triangle): There is 1 such upward triangle.

Total triangles pointing upwards = 6 + 3 + 1 = 10.

The number of triangles pointing up is 10.

(ii) Triangles pointing down:

• Size 1 (unit downward triangles): There are 3 unit downward triangles.

• No larger downward triangles exist in this figure.

Total downward triangles = 3

Total number of triangles = Triangles pointing up + Triangles pointing down = 10 + 3 = 13

There are total 13 triangles in the figure.
In simple words: Count small triangles of each size separately. Some small triangles join together to make bigger triangles, so you must count all sizes.

Exam Tip: Always count triangles by size - list how many unit triangles exist, then how many are made of 4 units, then how many are made of 9 units, and so on, based on how the figure is divided.

Question 10. In the adjoining figure, find the total number of squares.
Answer: The adjoining figure shows a 4 × 4 grid, that is, a large square divided into 16 unit squares by horizontal and vertical lines.

To find the total number of squares, we need to count squares of each possible size separately.

• Squares of size 1 × 1: There are 4 × 4 = 16 such squares.

• Squares of size 2 × 2: There are 3 × 3 = 9 such squares.

• Squares of size 3 × 3: There are 2 × 2 = 4 such squares.

• Squares of size 4 × 4: There is 1 × 1 = 1 such square (the whole figure).

Total number of squares = 16 + 9 + 4 + 1 = 30

There are total 30 squares in the figure.
In simple words: Count smallest squares first, then larger squares made from groups of smaller ones, then even larger squares, until you reach the biggest one.

Exam Tip: For an n × n grid, count squares systematically: n² squares of size 1 × 1, then (n-1)² of size 2 × 2, then (n-2)² of size 3 × 3, and so on - this method works for any grid size.