CBSE Class 8 Mathematics Practical Geometry Assignment Set 01

Question. If we add the number of REVOLUTIONS made by the hour hand and the minute hand in a day (24 hours), we would get:

(A) 25
(B) 26
(C) 72
(D) 84

Answer : B

Question. Dhruv's alarm clock gains 3 seconds every minute. Every night at 10:00 PM, he adjusts the clock to show the correct time and sets the alarm for 6:20 AM. What is the ACTUAL time when Dhruv's alarm clock rings in the morning?
(A) 5:36 AM
(B) 5:55 AM
(C) 5:20 AM
(D) 6:45 AM

Answer : B

Question. How many DIFFERENT sized squares can be drawn using the dots of the grid below as corners ?

(A) 4
(B) 3
(C) 2
(D) 1

Answer : B

Question. Given below are the steps of construction to construct a quadrilateral ABCD where AB = 5.6 cm, BC = 4.1 cm, CD = 4.4 cm, AD = 3.3 cm and ∠A = 75°. Which of the following steps is INCORRECT?
Step 1 : Draw AB = 5.6 cm and construct ∠BAX = 75°.
Step 2 : With A as centre and radius = 3.3 cm, cut off AD = 3.3 cm along AX.
Step 3 : Join BD. With D as centre and radius = 4.1 cm, draw an arc.
Step 4 : With B as centre and radius = 4.1 cm, draw an arc to cut the arc drawn in above step at C. Join BC, CD to obtain the required quadrilateral ABCD.
(A) Step 1 only
(B) Step 2 only
(C) Step 3 only
(D) Step 4 only
Answer : C

Question. Arrange the steps of construction while constructing a parallelogram ABCD, given that AB = 5 cm, BC = 4 cm and ∠B = 60°.
1. With A as centre and radius equal to 4 cm, draw an arc cutting AY at D.
2. At A, draw ∠YAB = 120°. [∴ A + B = 180°]
3. At B, draw ∠XBA = 60°.
4. Draw AB = 5 cm.
5. Join CD.
6. With B as centre and radius equal to 4 cm, drawn an arc cutting BX at C.
(A) 4, 3, 6, 2, 1, 5
(B) 4, 3, 2, 6, 5, 1
(C) 4, 6, 3, 1, 2, 5
(D) 4, 3, 6, 2, 5, 1
Answer : A

Question. To construct a quadrilateral ABCD, which of the following parts is necessary ?
(A) Length of AB
(B) Length of BC
(C) Measure of ∠A, ∠B and ∠C
(D) All of these
Answer : D

Question. Arrange the steps (i) to (iv) in correct order, while constructing a parallelogram ABCD,
given AB = 6 cm, AD = 4 cm, and diagonal BD = 3 cm.
Step 1 : Draw AB = 6 cm
Step (i) : Join AD and BD.
Step (ii) : With A as centre, draw an arc of radius 4 cm.
Step (iii) : With B and D as centres and with 4 cm and 6 cm as radii, respectively, draw arcs to cut each other at C.
Step (iv) : With B as centre, draw an arc of radius 3 cm to cut the arc drawn in step (ii) at point D.
Step 6 : Join CD and BC. ABCD is the required parallelogram.
(A) (ii), (iv), (iii), (i)
(B) (iii), (ii), (v), (iv)
(C) (ii), (iv), (i), (iii)
(D) None of these
Answer : C

Question. If AB||DC, AB = 7 cm, BC = 6 cm, AD = 6.5 cm and ∠B = 70°, then which figure can be constructed?
(A) Square
(B) Trapezium
(C) Rhombus
(D) Rectangle
Answer : B

Question. Given below are the steps of construction of a quadrilateral ABCD, where AB = 3.5 cm, BC = 6.5 cm, ∠A = 75°, ∠B = 105° and ∠C = 120°. Which of the following steps is INCORRECT?
Step 1 : Draw AB = 3.5 cm.
Step 2 : Draw ∠XAB = 75° at A and ∠ABY = 105° at B.
Step 3 : With B as centre and radius BC = 6.5 cm, draw an arc to intersect BY at C.
Step 4 : At C, draw ∠ADC = 120° such that CZ meets AX at D.
(A) Step 1 only
(B) Step 2 only
(C) Step 3 only
(D) Step 4 only
Answer : D

Question. To construct a kite, which of the following is necessary?
(A) Two adjacent unequal sides and included diagonal
(B) Two adjacent equal sides and included diagonal
(C) Length of opposite sides
(D) None of these
Answer : A

Question. Arrange the steps of construction while constructing a quadrilateral ABCD given AB = 5.1 cm, AD = 4 cm, BC = 2.5 cm, ∠A = 60° and ∠B = 85°.
Step 1 : With B as centre and radius 2.5 cm, cut off BC = 2.5 cm along BY.
Step 2 : Construct ∠XAB = 60° at A.
Step 3 : Join CD.
Step 4 : With A as centre and radius 4 cm, cut off AD = 4 cm along AX.
Step 5 : Draw AB = 5.1 cm.
Step 6 : Construct ∠ABY = 85° at B.
(A) 5, 2, 4, 1, 3, 6
(B) 5, 4, 2, 1, 6, 3
(C) 5, 2, 4, 6, 1, 3
(D) 5, 2, 4, 1, 6, 3
Answer : C

Question. Which of the following statements is true about the construction of a quadrilateral where
AB = 3 cm, BC = 5 cm, AC = 9 cm,
AD = 6 cm, CD = 2 cm?
(A) It is possible to draw the quadrilateral.
(B) It is not possible to draw the quadrilateral, since AD + DC < AC.
(C) It is possible to draw the quadrilateral, since AD + DC < AC
(D) None of these
Answer : B

Question. It is possible to construct a quadrilateral with the sufficient data (other than five simple cases), where less than ___ parts but some other relations between them are given.
(A) Four
(B) Five
(C) Three
(D) Two
Answer : B

Question. To construct a convex quadrilateral, which of the following cases is INCORRECT ?
(A) When the lengths of four sides and one diagonal are given.
(B) When the lengths of three sides and the two diagonals are given.
(C) When the lengths of four sides and one angle are given.
(D) When the lengths of two sides and two included angles are given.
Answer : D

Question. Which of the following steps is INCORRECT while constructing a rhombus ABCD, given that AC = 8 cm and BD = 6 cm?
Step 1 : Draw AC = 8 cm.
Step 2 : Draw PQ, the perpendicular of AC. PQ intersects AC at point O.
Step 3 : With O as centre and radius equal to 3 cm, drawn an arc cutting OP at D.
Step 4 : With O as centre and radius equal to 3 cm, draw another arc cutting OQ at B.
Step 5 : Join AB, BC, CD and DA.
(A) Step 2 only
(B) Step 3 only
(C) Step 4 only
(D) Both Step 2 and Step 5
Answer : A

Question. Which of the given properties of a parallelogram is necessary to construct it ?
(A) Opposite sides of a parallelogram
(B) Opposite angles of a parallelogram
(C) Diagonals of a parallelogram
(D) Both (A) and (B)
Answer : D

Question. Arrange the steps of construction while constructing a quadrilateral ABCD given AB = 5.3 cm, AD = 2.9 cm, ∠A = 70°, ∠B = 95° and ∠C = 85°.
Step 1 : With A as centre and radius 2.9 cm, draw an arc to cut AX at D.
Step 2 : At B, draw ∠ZBA = 95° so that ZB intersects YD at C.
Step 3 : Draw AB = 5.3 cm.
Step 4 : At D draw ∠ADY = 110°
Step 5 : At A draw ∠XAB = 70°.
(A) 3, 4, 2, 1, 5
(B) 3, 5, 1, 4, 2
(C) 3, 1, 5, 4, 2
(D) 3, 5, 4, 2, 1
Answer : B

Question. Match the following.

      P     Q      R     S
(A) 3      2      4      1
(B) 3      4      2      1
(C) 2      4      1      3
(D) 4      3      1      2
Answer : C

Q 1 Name the quadrilateral whose opposite sides are parallel.
Answer: A parallelogram is a quadrilateral whose opposite sides are parallel.
In simple words: A four-sided shape with parallel opposite sides is called a parallelogram.

Exam Tip: Remember that rectangles, squares, and rhombuses are also special types of parallelograms.

Q 2 Name the quadrilateral whose diagonals are equal and bisect each other at right angle.
Answer: A square is the quadrilateral whose diagonals are equal in length and bisect each other at \(90^\circ\).
In simple words: A square has equal diagonals that cross each other perpendicularly.

Exam Tip: A rhombus also has diagonals intersecting at right angles, but they are not equal in length.

Q 3 Why do we call square as a regular quadrilateral?
Answer: We call a square a regular quadrilateral because all its four interior angles are equal (\(90^\circ\)) and all its four sides are equal in length.
In simple words: A polygon with all sides and all angles equal is called a regular polygon.

Exam Tip: Regular means equiangular and equilateral.

Q 4 Construct the quadrilateral ABCD in which AB = 5 cm, BC = 4 cm, \(\angle A = 60^\circ\), \(\angle B = 105^\circ\) and \(\angle C = 105^\circ\).
Answer: 
Steps of Construction:
1. Draw line segment AB = 5 cm.
2. At point A, draw a ray making an angle of \(60^\circ\) with AB.
3. At point B, draw a ray making an angle of \(105^\circ\) with AB.
4. With B as center, draw an arc of radius 4 cm to cut the ray at point C.
5. At point C, draw a ray making an angle of \(105^\circ\) which intersects the ray from A at point D.
6. Join C and D to complete the required quadrilateral ABCD.
In simple words: Draw the base, construct the given angles at the vertices, cut off the side length, and close the shape.

Exam Tip: Always use a protractor or compass accurately for angle construction.

Q 5 The adjacent sides SP and PQ of a parallelogram PQRS are 4 cm each. State the measure of all the sides. What is another name of this figure?
Answer: 
In a parallelogram, opposite sides are equal.
Given adjacent sides SP = 4 cm and PQ = 4 cm.
Therefore, all sides are equal: PQ = QR = RS = SP = 4 cm.
Another name for this figure is a rhombus (or a square if interior angles are \(90^\circ\), but generally a parallelogram with all equal sides is a rhombus).
In simple words: When adjacent sides of a parallelogram are equal, all four sides become equal, forming a rhombus.

Exam Tip: A parallelogram with all equal sides is specifically a rhombus.

Q 6 The ratio of two adjacent sides of a parallelogram is 4:5. If its perimeter is 72 cm, find its adjacent sides.
Answer: 
Let the two adjacent sides be \(4x\) and \(5x\).
Perimeter of a parallelogram = \(2 \times (\text{sum of adjacent sides})\)
\(72 = 2 \times (4x + 5x)\)
\(72 = 2 \times 9x\)
\(72 = 18x\)
\(x = \frac{72}{18} = 4 \text{ cm}\)
First side = \(4x = 4 \times 4 = 16 \text{ cm}\)
Second side = \(5x = 5 \times 4 = 20 \text{ cm}\)
In simple words: Use the perimeter formula to find the value of the ratio variable, then multiply to get side lengths.

Exam Tip: Perimeter of a parallelogram is calculated similar to a rectangle: \(2(l + b)\).

Q 7 The park in a town is made in the form of a kite. Its perimeter is 90 metres and one side is 10m more than other side. What are the lengths of other sides?
Answer: 
A kite has two pairs of equal adjacent sides.
Let one pair of unequal sides be \(x\) and \(x + 10\).
Perimeter = \(2 \times (x + (x + 10)) = 90\)
\(2(2x + 10) = 90\)
\(2x + 10 = 45\)
\(2x = 35\)
\(x = 17.5 \text{ m}\)
The sides are 17.5 m, 17.5 m, 27.5 m, and 27.5 m.
In simple words: A kite has two pairs of equal sides, so the perimeter formula includes two of each distinct side length.

Q 8 Construct the trapezium ABCD in which \(AB \parallel CD\), AB = 8 cm, BC = 6 cm, CD = 4 cm and \(\angle B = 60^\circ\).
Answer: 
Steps of Construction:
1. Draw line segment AB = 8 cm.
2. At point B, construct an angle of \(\angle B = 60^\circ\).
3. With center B and radius 6 cm, draw an arc intersecting the ray at point C.
4. Through C, draw a line parallel to AB.
5. With center C and radius 4 cm, draw an arc cutting the parallel line at point D.
6. Join A to D to complete trapezium ABCD.
In simple words: Construct the base and side with the given angle, draw a parallel line, and cut off the top side length.

Exam Tip: Use alternate interior angles or set squares to draw parallel lines accurately.

Q 9 Construct the quadrilateral ABCD in which AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and AC = 7 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 7 cm.
2. With A as center and radius 4.5 cm, draw an arc.
3. With C as center and radius 5.5 cm, draw an arc intersecting the previous arc at B. Join AB and CB.
4. With A as center and radius 6 cm, draw an arc on the other side of AC.
5. With C as center and radius 4 cm, draw an arc intersecting the previous arc at D. Join AD and CD.
6. ABCD is the required quadrilateral.
In simple words: Construct two triangles sharing a common diagonal AC.

Exam Tip: When four sides and one diagonal are given, always start by constructing the triangle that has the diagonal as its base.

Q 10 Construct the quadrilateral PQRS where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm.
Answer: 
Steps of Construction:
1. Draw diagonal PR = 7 cm.
2. With P as center and radius 4 cm, draw an arc.
3. With R as center and radius 6 cm, draw an arc intersecting at Q. Join PQ and RQ.
4. With P as center and radius 5.5 cm, draw an arc on the other side of PR.
5. With R as center and radius 5 cm, draw an arc intersecting at S. Join PS and RS.
6. PQRS is the required quadrilateral.
In simple words: Split the quadrilateral into two triangles using diagonal PR and draw them side by side.

Exam Tip: Label all vertices correctly according to the given cyclic order.

Q 11 Construct Quadrilateral ABCD in which AB = 3.5 cm, BC = 4 cm, CD = 5 cm, \(\angle B = 45^\circ\) and \(\angle C = 150^\circ\).
Answer: 
Steps of Construction:
1. Draw line segment BC = 4 cm.
2. At point B, construct an angle of \(45^\circ\) and cut off AB = 3.5 cm.
3. At point C, construct an angle of \(150^\circ\) and cut off CD = 5 cm.
4. Join points A and D.
5. ABCD is the required quadrilateral.
In simple words: Draw the middle base BC, construct the given angles at B and C, mark the side lengths, and join the endpoints.

Exam Tip: An angle of \(150^\circ\) can be constructed by combining \(90^\circ\) and \(60^\circ\) or using a protractor.

Q 12 Construct the quadrilateral TRUE in which TR = 3.5 cm, RU = 3 cm, UE = 4 cm, \(\angle R = 75^\circ\) and \(\angle U = 120^\circ\).
Answer: 
Steps of Construction:
1. Draw line segment RU = 3 cm.
2. At point R, construct an angle of \(75^\circ\) and cut off TR = 3.5 cm.
3. At point U, construct an angle of \(120^\circ\) and cut off UE = 4 cm.
4. Join points T and E.
5. TRUE is the required quadrilateral.
In simple words: Start with the central side RU, build the given angles at each end, mark off the adjacent sides, and connect them.

Exam Tip: \(75^\circ\) is constructed by bisecting the angle between \(60^\circ\) and \(90^\circ\).

Q 13 Construct the parallelogram ABCD with AB = 3.5 cm, BC = 4 cm and AC = 6.5 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 6.5 cm.
2. With A as center and radius 3.5 cm (equal to opposite side CD/AB), draw an arc.
3. With C as center and radius 4 cm (equal to opposite side BC), draw an arc intersecting at B. Join AB and CB.
4. With A as center and radius 4 cm, draw an arc on the other side of AC.
5. With C as center and radius 3.5 cm, draw an arc intersecting at D. Join AD and CD.
6. ABCD is the required parallelogram.
In simple words: Construct triangle ABC first, then use opposite side properties to locate vertex D.

Exam Tip: Opposite sides of a parallelogram are equal in length.

Q 14 Construct a rhombus with side 4.5 cm and one diagonal 6 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 6 cm.
2. With A as center and radius 4.5 cm, draw an arc above AC and another below AC.
3. With C as center and radius 4.5 cm, draw arcs to intersect the previous arcs at B and D.
4. Join AB, BC, CD, and DA.
5. ABCD is the required rhombus.
In simple words: A rhombus has all four sides equal, so use the diagonal as the base of two congruent isosceles triangles.

Exam Tip: All four sides of a rhombus are equal in length.

Q 15 Construct the quadrilateral ABCD with AB = 4 cm, BC = 6 cm, CD = 5.5 cm, AD = 5 cm and AC = 8 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 8 cm.
2. With A as center and radius 4 cm, and C as center and radius 6 cm, draw arcs intersecting at B. Join AB and BC.
3. With A as center and radius 5 cm, and C as center and radius 5.5 cm, draw arcs intersecting at D. Join AD and CD.
4. ABCD is the required quadrilateral.
In simple words: Construct triangles ABC and ADC sharing the common diagonal AC.

Exam Tip: Check that the sum of any two sides of the triangle is greater than the third side before drawing arcs.

Q 16 Construct the quadrilateral HOPE with HO = 4.5 cm, OP = 4 cm, PE = 6.5 cm, EH = 3 cm and OE = 6.5 cm.
Answer: 
Steps of Construction:
1. Draw diagonal OE = 6.5 cm.
2. With O as center and radius 4.5 cm, and E as center and radius 3 cm, draw arcs intersecting at H. Join OH and EH.
3. With O as center and radius 4 cm, and E as center and radius 6.5 cm, draw arcs intersecting at P. Join OP and EP.
4. HOPE is the required quadrilateral.
In simple words: Use diagonal OE to construct triangles OHE and OPE.

Exam Tip: Verify all side lengths match the given dimensions precisely.

Q 17 Construct the quadrilateral PQRS in which PQ = 4 cm, QR = 3 cm, PS = 2.5 cm, PR = 4.5 cm and QS = 4 cm.
Answer: 
Steps of Construction:
1. Draw diagonal PR = 4.5 cm.
2. With P as center and radius 4 cm, and R as center and radius 3 cm, draw arcs intersecting at Q. Join PQ and QR.
3. To locate S, we use PR = 4.5 cm, PS = 2.5 cm, and QS = 4 cm. Construct triangle PQS with sides PQ = 4 cm, PS = 2.5 cm, and QS = 4 cm.
4. Join RS to complete quadrilateral PQRS.
In simple words: Construct triangle PQR first, then use diagonals and side lengths to find S.

Q 18 ABCD is a trapezium with \(AB \parallel CD\), and \(\angle A = 50^\circ\) and \(\angle B = 50^\circ\). Prove that (i) BC = DA (ii) \(\angle C = \angle D\) and find the measurement of \(\angle C\).
Answer: 
(i) Since \(AB \parallel CD\) and AD, BC are transversal lines, and given \(\angle A = \angle B = 50^\circ\), the base angles are equal, which implies the trapezium is isosceles. Therefore, non-parallel sides are equal, so BC = DA.
(ii) In an isosceles trapezium, base angles are equal and top angles are equal. Thus, \(\angle C = \angle D\).
Since consecutive interior angles between parallel lines sum to \(180^\circ\):
\(\angle A + \angle D = 180^\circ\)
\(50^\circ + \angle D = 180^\circ\)
\(\angle D = 130^\circ\)
Therefore, \(\angle C = \angle D = 130^\circ\).
In simple words: Equal base angles mean the trapezium is symmetric (isosceles), so opposite non-parallel sides and top angles are equal.

Exam Tip: Consecutive interior angles add up to \(180^\circ\) when lines are parallel.

Q 19 The perimeter of a parallelogram is 140 cm. If one of the sides is greater than the other by 20 cm, find the lengths of all the sides of the parallelogram.
Answer: 
Let one side of the parallelogram be \(x \text{ cm}\).
The adjacent side = \(x + 20 \text{ cm}\).
Perimeter = \(2 \times (\text{sum of adjacent sides}) = 140\)
\(2(x + x + 20) = 140\)
\(2(2x + 20) = 140\)
\(2x + 20 = 70\)
\(2x = 50\)
\(x = 25 \text{ cm}\)
First side = 25 cm
Second side = \(25 + 20 = 45 \text{ cm}\)
The lengths of all four sides are 25 cm, 45 cm, 25 cm, and 45 cm.
In simple words: Set up an algebraic equation for the perimeter, solve for the smaller side, and find its adjacent larger side.

Exam Tip: Opposite sides of a parallelogram are equal, so there are two sides of length 25 cm and two of 45 cm.

Q 20 ABCD is a trapezium in which \(AB \parallel DC\) and AD = BC. If CE is drawn parallel to AD, meeting AB at E, prove the following: (i) AECD is a parallelogram. (ii) AD = EC (iii) \(\angle CEB\) is an isosceles triangle.
Answer: 
(i) In quadrilateral AECD, \(AE \parallel DC\) (since \(AB \parallel DC\)) and \(AD \parallel CE\) (given). Since both pairs of opposite sides are parallel, AECD is a parallelogram.
(ii) Since AECD is a parallelogram, opposite sides are equal. Therefore, AD = EC.
(iii) We are given AD = BC. Since AD = EC, it follows that BC = EC. In triangle CEB, since side BC = side EC, the triangle has two equal sides, making \(\triangle CEB\) an isosceles triangle.
In simple words: By constructing a parallel line, we form a parallelogram and prove that the remaining triangle has two equal sides.

Exam Tip: Properties of parallelograms are extremely useful for proving line segment equalities in geometry proofs.

Q 21 Construct the square READ with RE = 5.1 cm.
Answer: 
Steps of Construction:
1. Draw line segment RE = 5.1 cm.
2. At points R and E, construct angles of \(90^\circ\).
3. With centers R and E, draw arcs of radius 5.1 cm along the perpendicular rays to mark points A and D.
4. Join A and D.
5. READ is the required square.
In simple words: Draw a base of 5.1 cm, erect right angles at both ends, cut off 5.1 cm on each, and join the top.

Exam Tip: All sides of a square are equal and all interior angles are \(90^\circ\).

Q 22 Construct a rhombus ABCD where AC = 5.2 cm and BD = 6.4 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 5.2 cm.
2. Draw the perpendicular bisector of AC, intersecting AC at point O.
3. From center O, cut off arcs of length \(\frac{6.4}{2} = 3.2 \text{ cm}\) on both sides of the perpendicular bisector to get points B and D.
4. Join AB, BC, CD, and DA.
5. ABCD is the required rhombus.
In simple words: Use the property that diagonals of a rhombus bisect each other at right angles.

Exam Tip: Remember to divide the second diagonal length by 2 when measuring from the intersection point.

Q 23 Construct a rectangle ABCD with AB = 5 cm and BC = 4 cm.
Answer: 
Steps of Construction:
1. Draw line segment AB = 5 cm.
2. At points A and B, construct angles of \(90^\circ\).
3. With centers A and B, draw arcs of radius 4 cm along the perpendicular rays to locate points D and C respectively.
4. Join D and C.
5. ABCD is the required rectangle.
In simple words: Draw the base, erect \(90^\circ\) angles at the ends, mark the breadth, and close the top.

Exam Tip: Opposite sides of a rectangle are equal and all angles are \(90^\circ\).

Q 24 Construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, \(\angle A = 55^\circ\), \(\angle B = 110^\circ\) and \(\angle D = 90^\circ\).
Answer: 
First, find the fourth angle \(\angle C\). The sum of interior angles of a quadrilateral is \(360^\circ\):
\(\angle A + \angle B + \angle C + \angle D = 360^\circ\)
\(55^\circ + 110^\circ + \angle C + 90^\circ = 360^\circ\)
\(255^\circ + \angle C = 360^\circ\)
\(\angle C = 360^\circ - 255^\circ = 105^\circ\)
Steps of Construction:
1. Draw line segment AB = 6 cm.
2. At A, make \(\angle A = 55^\circ\). At B, make \(\angle B = 110^\circ\).
3. With B as center and radius 5 cm, cut an arc on the ray from B to locate point C.
4. At C, construct \(\angle C = 105^\circ\) to intersect the ray from A at point D.
5. ABCD is the required quadrilateral.
In simple words: Calculate the missing angle using angle sum property, then construct using base AB and adjacent angles.

Exam Tip: Always calculate any missing angles using the \(360^\circ\) quadrilateral angle sum property before starting construction.

Most Important Questions

Q 1 Is it possible to construct a quadrilateral ABCD in which AB = 3 cm, CD = 3 cm, DA = 7.5 cm, AC = 8 cm and BD = 4 cm? If not, give reason.
Answer: 
Consider triangle ADC with sides AD = 7.5 cm, CD = 3 cm, and AC = 8 cm.
Check the triangle inequality theorem (sum of any two sides must be greater than the third side):
\(CD + CD = 3 + 7.5 = 10.5 > 8\) (Valid)
\(CD + AC = 3 + 8 = 11 > 7.5\) (Valid)
Now consider triangle ABC with AB = 3 cm, BC (unknown), AC = 8 cm. Wait, let us check triangle ABD with AB = 3 cm, BD = 4 cm, AD = 7.5 cm:
\(AB + BD = 3 + 4 = 7\)
Here, \(AB + BD = 7\), which is NOT greater than AD (\(7.5\)).
Therefore, triangle ABD cannot be formed.
In simple words: No, because the sum of two sides (3 cm + 4 cm = 7 cm) is less than the third side (7.5 cm), violating the triangle inequality property.

Exam Tip: Always verify triangle inequalities for side lengths before concluding construction feasibility.

Q 2 Is it possible to construct a quadrilateral ABCD in which AB = 5 cm, BC = 7.5 cm, \(\angle A = 80^\circ\), \(\angle B = 140^\circ\), \(\angle C = 145^\circ\)? If not, give reason.
Answer: 
Let us find the sum of the given interior angles of the quadrilateral:
\(\angle A + \angle B + \angle C = 80^\circ + 140^\circ + 145^\circ = 365^\circ\)
Since the sum of three interior angles alone exceeds \(360^\circ\), it is impossible to construct such a quadrilateral.
In simple words: No, because the sum of the given three angles is already \(365^\circ\), which is greater than the total sum of all four angles of a quadrilateral (\(360^\circ\)).

Exam Tip: The sum of all interior angles of any quadrilateral is always exactly \(360^\circ\).

Q 3 Construct a quadrilateral ABCD in which AB = 4.4 cm, BC = 4 cm, CD = 6.4 cm, DA = 2.8 cm and BD = 6.6 cm.?
Answer: 
Steps of Construction:
1. Draw diagonal BD = 6.6 cm.
2. With B as center and radius 4 cm, and D as center and radius 2.8 cm, draw arcs intersecting at C. Join BC and DC.
3. With B as center and radius 4.4 cm, and D as center and radius 6.4 cm, draw arcs intersecting at A. Join AB and AD.
4. ABCD is the required quadrilateral.
In simple words: Use diagonal BD as the shared base to construct triangles BCD and BAD.

Exam Tip: Choosing the diagonal that forms valid triangles on both sides ensures successful construction.

Q 4 Construct a parallelogram ABCD where AB = 3.6 cm, BC = 4.2 cm and AC = 6.5 cm.?
Answer: 
Steps of Construction:
1. Draw diagonal AC = 6.5 cm.
2. With A as center and radius 3.6 cm, and C as center and radius 4.2 cm, draw arcs intersecting at B. Join AB and BC.
3. With A as center and radius 4.2 cm, and C as center and radius 3.6 cm, draw arcs intersecting at D on the opposite side of AC. Join AD and CD.
4. ABCD is the required parallelogram.
In simple words: Construct triangle ABC, then use opposite sides to complete the parallelogram.

Exam Tip: Opposite sides of a parallelogram are equal: AD = BC and CD = AB.

Q 5 Construct a rhombus with side 4.5 cm and one diagonal 6 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 6 cm.
2. With A as center and radius 4.5 cm, draw arcs above and below AC.
3. With C as center and radius 4.5 cm, draw arcs intersecting the previous arcs at B and D.
4. Join AB, BC, CD, and DA.
5. ABCD is the required rhombus.
In simple words: Draw the diagonal and use equal side lengths to swing intersecting arcs for the other two vertices.

Exam Tip: A rhombus has all four sides equal in length.

Q 6 Construct a quadrilateral ABCD in which AB = 5 cm, BC = 4 cm, AD = 3 cm, CD = 6 cm and BD = 5 cm.
Answer: 
Steps of Construction:
1. Draw diagonal BD = 5 cm.
2. With B as center and radius 4 cm, and D as center and radius 6 cm, draw arcs intersecting at C. Join BC and DC.
3. With B as center and radius 5 cm, and D as center and radius 3 cm, draw arcs intersecting at A. Join AB and AD.
4. ABCD is the required quadrilateral.
In simple words: Construct two triangles, ABD and CBD, using diagonal BD as their common base.

Q 7 Construct a quadrilateral ABCD in which AB = BC = 3 cm, AD = 5 cm, \(\angle A = 90^\circ\) and \(\angle B = 105^\circ\).
Answer: 
Steps of Construction:
1. Draw line segment AB = 3 cm.
2. At point A, construct \(\angle A = 90^\circ\) and cut off AD = 5 cm.
3. At point B, construct \(\angle B = 105^\circ\) and cut off BC = 3 cm.
4. Join points C and D.
5. ABCD is the required quadrilateral.
In simple words: Draw base AB, erect the given angles at A and B, mark off side lengths AD and BC, and join C to D.

Exam Tip: Construct \(90^\circ\) at A and \(105^\circ\) at B using standard compass and ruler techniques.

Q 8 Construct a rectangle with sides 4.5 cm and 6 cm.
Answer: 
Steps of Construction:
1. Draw line segment AB = 6 cm.
2. At points A and B, construct \(90^\circ\) angles.
3. With centers A and B, draw arcs of radius 4.5 cm along the perpendicular lines to locate points D and C.
4. Join D and C.
5. ABCD is the required rectangle.
In simple words: Draw the length as base, construct right angles at both ends, mark the breadth, and close the shape.

Exam Tip: Adjacent angles of a rectangle are always \(90^\circ\).

Q 9 Construct a quadrilateral ABCD in which AB = 7 cm, AD = 6 cm, AC = 7 cm, BD = 7.5 cm and BC = 5 cm.
Answer: 
Steps of Construction:
1. Draw diagonal AC = 7 cm.
2. With A as center and radius 7 cm, and C as center and radius 5 cm, draw arcs intersecting at B. Join AB and BC.
3. With A as center and radius 6 cm, and B as center and radius 7.5 cm (using diagonal BD and vertex A/B relations), draw arcs to locate D.
4. Join AD and CD.
5. ABCD is the required quadrilateral.
In simple words: Construct triangle ABC first, then use remaining side and diagonal lengths to locate D.

Exam Tip: Keep track of diagonal lengths when intersecting arcs for opposite vertices.

Q 10 Construct a quadrilateral ABCD in which AB = 5.5 cm, AD = 4.4 cm, CD = 6.5 cm, AC = 6.5 cm and BD = 7.1 cm.?
Answer: 
Steps of Construction:
1. Draw diagonal AC = 6.5 cm.
2. With A as center and radius 5.5 cm, and C as center and radius 6.5 cm, draw arcs intersecting at B. Join AB and BC.
3. With A as center and radius 4.4 cm, and B as center and radius 7.1 cm (using BD), draw arcs intersecting at D. Join AD and CD.
4. ABCD is the required quadrilateral.
In simple words: Construct triangle ABC, then use side AD and diagonal BD to pinpoint vertex D.

Exam Tip: Ensure accurate arc intersections when working with multiple given diagonals.

Q 11 Construct a quadrilateral ABCD in which AB = 5.4 cm, BC = 2.5 cm, CD = 4 cm, AC = 6.5 cm and BD = 5 cm.?
Answer: 
Steps of Construction:
1. Draw diagonal AC = 6.5 cm.
2. With A as center and radius 5.4 cm, and C as center and radius 2.5 cm, draw arcs intersecting at B. Join AB and BC.
3. To locate D, use triangle BCD or ACD with given diagonal BD = 5 cm and CD = 4 cm. With C as center and radius 4 cm, and B as center and radius 5 cm (BD), draw arcs to intersect at D.
4. Join AD and CD.
5. ABCD is the required quadrilateral.
In simple words: Build triangle ABC on diagonal AC, then locate D using sides CD and BD.

Exam Tip: Label every intersection clearly as you progress through the construction steps.

Q 12 Construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, \(\angle A = 55^\circ\), \(\angle B = 110^\circ\) and \(\angle D = 90^\circ\).
Answer: 
First, find angle C using the angle sum property of a quadrilateral: sum = \(360^\circ\).
\(\angle C = 360^\circ - (55^\circ + 110^\circ + 90^\circ) = 360^\circ - 255^\circ = 105^\circ\).
Steps of Construction:
1. Draw line segment AB = 6 cm.
2. Construct \(\angle A = 55^\circ\) and \(\angle B = 110^\circ\).
3. With B as center and radius 5 cm, cut off BC = 5 cm along the ray from B.
4. At C, construct \(\angle C = 105^\circ\) to intersect the ray from A at point D.
5. ABCD is the required quadrilateral.
In simple words: Calculate the missing angle C first, then construct the quadrilateral from base AB.

Exam Tip: Always find any unknown interior angle before starting angle-side-angle constructions.

Q 13 Construct a quadrilateral PQRS in which \(\angle Q = 45^\circ\), \(\angle R = 90^\circ\), QR = 5 cm, PQ = 4 cm and RS = 3 cm.
Answer: 
Steps of Construction:
1. Draw line segment QR = 5 cm.
2. At point Q, construct \(\angle Q = 45^\circ\) and cut off PQ = 4 cm.
3. At point R, construct \(\angle R = 90^\circ\) and cut off RS = 3 cm.
4. Join points P and S.
5. PQRS is the required quadrilateral.
In simple words: Draw the middle base QR, construct the given angles at Q and R, mark off side lengths, and connect the endpoints.

Exam Tip: Construct \(45^\circ\) by bisecting a \(90^\circ\) angle.

Q 14 Construct a quadrilateral ABCD in which AB = 3.6 cm, BC = 5.5 cm, CD = 5 cm, angle B = 125 and angle C = 80?
Answer: 
Steps of Construction:
1. Draw line segment BC = 5.5 cm.
2. At point B, construct an angle of \(125^\circ\) and cut off AB = 3.6 cm.
3. At point C, construct an angle of \(80^\circ\) and cut off CD = 5 cm.
4. Join points A and D.
5. ABCD is the required quadrilateral.
In simple words: Use BC as the base, build the given angles at B and C, mark the side lengths, and join the free ends.

Exam Tip: Use a protractor to accurately measure non-standard angles like \(125^\circ\) and \(80^\circ\).

Q 15 Construct a quadrilateral ABCD in which AB = 5.1 cm, AD = 4 cm, BC = 2.5 cm, angle A = 60 and angle B = 85?
Answer: 
Steps of Construction:
1. Draw line segment AB = 5.1 cm.
2. At point A, construct an angle of \(60^\circ\) and cut off AD = 4 cm.
3. At point B, construct an angle of \(85^\circ\) and cut off BC = 2.5 cm.
4. Join points D and C.
5. ABCD is the required quadrilateral.
In simple words: Draw base AB, build the given angles at A and B, mark off sides AD and BC, and join D to C.

Exam Tip: Ensure angle measurements are marked correctly from the baseline segment.

Q 16 Construct a quadrilateral ABCD in which AB = 3.5 cm, BC = 6.5 cm, \(\angle A = 75^\circ\) and \(\angle B = 105^\circ\) and \(\angle C = 120^\circ\)?
Answer: 
Steps of Construction:
1. Draw line segment AB = 3.5 cm.
2. At point B, construct \(\angle B = 105^\circ\) and cut off BC = 6.5 cm.
3. At point C, construct \(\angle C = 120^\circ\).
4. At point A, construct \(\angle A = 75^\circ\) such that its ray intersects the ray from C at point D.
5. ABCD is the required quadrilateral.
In simple words: Start with base AB, construct angles at A, B, and C, mark side BC, and find vertex D at the intersection of rays.

Exam Tip: Make sure angle rays are extended sufficiently so they intersect to form vertex D.

Q 17 Construct a quadrilateral ABCD in which AB = 5.3 cm, AD = 2.9 cm, \(\angle A = 70^\circ\) and \(\angle B = 95^\circ\) and \(\angle C = 85^\circ\)?
Answer: 
First, find the fourth angle \(\angle D\):
Sum of angles = \(360^\circ\)
\(\angle A + \angle B + \angle C + \angle D = 360^\circ\)
\(70^\circ + 95^\circ + 85^\circ + \angle D = 360^\circ\)
\(250^\circ + \angle D = 360^\circ\)
\(\angle D = 110^\circ\)
Steps of Construction:
1. Draw line segment AB = 5.3 cm.
2. At point A, construct \(\angle A = 70^\circ\) and cut off AD = 2.9 cm.
3. At point B, construct \(\angle B = 95^\circ\).
4. At point D, construct \(\angle D = 110^\circ\) to intersect the ray from B at point C.
5. ABCD is the required quadrilateral.
In simple words: Determine the missing angle first, then construct using base AB, side AD, and relevant angles.

Q 18 Construct a quadrilateral in which QR = 7.5 cm, RP = PS = 6 cm, RS = 5 cm and QS = 10 cm.
Answer: 
Steps of Construction:
1. Draw diagonal QS = 10 cm.
2. With Q as center and radius QR = 7.5 cm, and S as center and radius RS = 5 cm, draw arcs intersecting at R. Join QR and SR.
3. With S as center and radius PS = 6 cm, and R as center and radius RP = 6 cm, draw arcs intersecting at P. Join SP and RP.
4. PQRS is the required quadrilateral.
In simple words: Use diagonal QS as the base to construct triangles QRS and QPS (or RPS).

Exam Tip: Double-check compass measurements when constructing triangles with equal side lengths.

Q 19 Construct a trapezium ABCD in which \(AB \parallel CD\), AB = 8 cm, BC = 6.0 cm and CD = 4 cm and \(\angle B = 60^\circ\).
Answer: 
Steps of Construction:
1. Draw line segment AB = 8 cm.
2. At point B, construct \(\angle B = 60^\circ\).
3. With center B and radius 6 cm, cut an arc on the ray at point C.
4. Through C, draw a line parallel to AB.
5. With center C and radius 4 cm (length of CD), cut an arc on the parallel line to locate point D.
6. Join A and D to complete trapezium ABCD.
In simple words: Construct base AB, build the angle and side at B, draw a parallel line from C, and mark CD.

Exam Tip: Use set squares or alternate angles to ensure CD is perfectly parallel to AB.

Q 20 Construct a parallelogram whose two sides and one angle are 4 cm, 5.5 cm and \(70^\circ\) respectively.
Answer: 
Steps of Construction:
1. Draw line segment AB = 5.5 cm.
2. At point A, construct an angle of \(70^\circ\).
3. Along the ray from A, cut off AD = 4 cm.
4. With center D and radius 5.5 cm, draw an arc.
5. With center B and radius 4 cm, draw an arc intersecting the previous arc at point C.
6. Join D to C and B to C.
7. ABCD is the required parallelogram.
In simple words: Construct the base, the given angle at one vertex, mark the adjacent side, and use opposite side lengths to complete the parallelogram.

Exam Tip: Opposite sides of a parallelogram are equal in length, making it easy to finish using arcs from vertices B and D.