RD Sharma Solutions Class 6 Maths Chapter 10 Basic Geomatrical Concepts

Exercise 10.1

Question 1:  Make three points in your notebook and name them. 

Solution 1:   Three points are P, Q and R as given below:

Question 2:  Draw a line in your notebook and name it using a small letter of the alphabet. 

Solution 2:  The line RS is drawn and is named as ‘s’

Question 3:  Draw a line in your notebook and name it by taking any two points on it. 

Solution 3:  Make a line and name the points are RS.

Question 11:  Is it ever possible for exactly one line to pass through three points? 

Solution 11:   Yes, It is possible for a line to pass through three points, but only if the points lie on a straight line.

Question 12:   Explain why it is not possible for a line to have a mid-point. 

Solution 12:   The length of the line goes on indefinitely in both directions, so a line has no midpoint and only a line segment can have a midpoint.

Question 13:   Mark three non-collinear points A, B and C in your note book. Draw lines through these points taking two at a time. Name these lines. How many such different lines can be drawn? 

Solution 13:

Given three collinear points are A, B and C.

As we know that three lines namely AB, BC and CA can be drawn using these points.

Question 14:   Coplanar points are the points that are in the same plane. Thus,

(i) Can 150 points be coplanar?

(ii) Can 3 points be non-coplanar? 

Solution 14:

(i) Yes, as we know that the group of points which lie in the same plane are coplanar points.

Therefore, 150 points can be coplanar.

(ii) No, 3 points can be coplanar as we can have plane which contains 3 points.

Therefore, 3 points cannot be non-coplanar.

Question 15:   Using a ruler, check whether the following points given in Fig. 10.30 are collinear or not:

(i) D, A and C

(ii) A, B and C

(iii) A, B and E

(iv) B, C and E

Solution 15:

(i) D, A and C: are collinear.

(ii) A, B and C: are non-collinear.

(iii) A, B and E: are collinear.

(iv) B, C and E: are non-collinear.

Question 16:  Lines p, q are coplanar. So are the lines p, r. Can we conclude that the lines p, q, r are coplanar? 

Solution 16:   Yes, The lines p, q and r are coplanar.

Question 17:  Give three examples each of:

(i) intersecting lines

(ii) parallel lines from your environment. 

Solution 17: 

Question 18:  From Fig. 10.21, write

(i) All pairs of parallel lines.

(ii) All pairs of intersecting lines.

(iii) Lines whose point of intersection is I.

(iv) Lines whose point of intersection is D.

(v) Lines whose point of intersection is E.

(vi) Lines whose point of intersection is A.

(vii) Collinear points. 

Solution 18:

(i) All pairs of parallel lines: [L,M], [M,N], [L,N].

(ii) All pairs of intersecting line:. [L,P], [M,P], [N, P]; [L, R]; [M, R]; [N, R]; [P, R]; [L, Q]; [M, Q]; [N, Q]; [Q, P]; [Q, R].

(iii) Lines whose point of intersection is I: M and P.

(iv) Lines whose point of intersection is D: L and R.

(v) Lines whose point of intersection is E: M and R.

(vi) Lines whose point of intersection is A. L and Q.

(vii) Collinear points: [G, A, B, C] [D, E, J, F]; [G, H, I, J, K]; [A, H, D]; [B, I, E]; [C, F, K].

Question 19:  From Fig. 10.22, write concurrent lines and their points of concurrence.

Solution 19:

The three or more lines which across the same point are concurrent lines.

Here, the lines n, q and l are concurrent with point A as the point of concurrence.

m, q and p are concurrent with point B as the point of concurrence.

Question 20:   Mark four points A, B, C and D in your notebook such that no three of them are collinear. Draw all the lines which join them in pairs as shown in Fig. 10.23.

(i) How many such lines can be drawn?

(ii) Write the names of these lines.

(iii) Name the lines which are concurrent at A. 

Solution 20:   

(i) Through the points A, B, C and D we can draw six lines.

(ii) AB, BC, CD, AD, BD and AC are the names of these lines.

(iii) AC, AB and AD are the lines which are concurrent at A.

Question 21:   What is the maximum number of points of intersection of three lines in a plane? What is the minimum number? 

Solution 21:   The maximum number of points of intersection of three lines in a plane are three.

Question 22:  With the help of a figure, find the maximum and minimum number of points of intersection of four lines in a plane. 

Solution 22:   The maximum six numbers of points of intersection of four lines in a plane.

Question 23:  Lines p, q and r are concurrent. Also, lines p, r and s are concurrent. Draw a figure and state whether lines p, q, r and s are concurrent or not. 

Solution 23:

The lines p, q and r intersect at the point O. So, the lines p, r and s are concurrent.

The lines p, r, s and p, q, r intersect at O.

Therefore, the lines p, q, r and s are concurrent intersecting at the point O.

Question 24:  Lines p, q and r are concurrent. Also lines p, s and t are concurrent. Is it always true that the lines q, r and s will be concurrent? Is it always true for lines q, r and t? 

Solution 24:   The lines p, q and r intersect at the point O and are concurrent.

The lines p, s and t intersect at the common point and are concurrent. It is not always true that the lines q, r and s or q, r and t are concurrent.

Question 25:   Fill in the blanks in the following statements using suitable words:

(i) A page of a book is a physical example of a ….

(ii) An inkpot has both ……. Surfaces

(iii) Two lines in a plane are either ……. or are …. 

Solution 25:

(i) A page of a book is a physical example of a plane.

(ii) An inkpot has both curved and plane surfaces.

(iii) Two lines in a plane are either parallel or are intersecting.

Question 26:  State which of the following statements are true (T) and which are false (F):

(i) Point has a size because we can see it as a thick dot on paper.

(ii) By lines in geometry, we mean only straight lines.

(iii) Two lines in a plane always intersect in a point.

(iv) Any plane through a vertical line is vertical.

(v) Any plane through a horizontal line is horizontal.

(vi) There cannot be a horizontal line in a vertical plane.

(vii) All lines in a horizontal plane are horizontal.

(viii) Two lines in a plane always intersect in a point.

(ix) If two lines intersect at a point P, then P is called the point of concurrence of the two lines.

(x) If two lines intersect at a point P, then P is called the point of intersection of the two lines.

(xi) If A, B, C and D are collinear points D, P and Q are collinear, then points A, B, C, D, P and Q are always collinear.

(xii) Two different lines can be drawn passing through two given points.

(xiii) Through a given point only one line can be drawn.

(xiv) Four points are collinear if any three of them lie on the same line.

(xv) The maximum number of points of intersection of three lines is three.

(xvi) The maximum number of points of intersection of three lines is one. 

Solution 26: