Practice Set 5.2

- Find the coordinates of point P if P divides the line segment joining the points…
- In each of the following examples find the co-ordinates of point A which divides…
- Find the ratio in which point T(-1, 6)divides the line segment joining the pointsP(-3,…
- Point P is the centre of the circle and AB is a diameter. Find the coordinates of point…
- Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) andB(1, 2).…
- Find the coordinates of midpoint of the segment joining the points (22,20) and (0, 16).…
- Find the centroids of the triangles whose vertices are given below. (1) (-7, 6), (2,…
- In ΔABC, G (-4, -7) is the centroid. If A (-14, -19) and B(3, 5) then find…
- A(h, -6), B(2, 3) and C(-6, k) are the co-ordinates of vertices of a triangle whose…
- Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7)…
- If A (-14, -10), B(6, -2) is given, find the coordinates of the points which divide…
- If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide…

###### Practice Set 5.2

Find the coordinates of point P if P divides the line segment joining the points A(-1,7) and B(4,-3) in the ratio 2:3.

Answer:

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

In the given question

x = = 1

y =

y = 3

Hence the coordinates of the point are (1,3).

Question 2.

In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a:b.

(1) P(-3, 7), Q(1, -4), a:b = 2:1

(2) P(-2, -5), Q(4, 3), a:b = 3:4

(3) P(2, 6), Q(-4, 1), a:b = 1:2

Answer:

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

Where m and n is defined as the ratio in which the line segments are divided

1. x =

x =

y =

y =

2. x =

X =

y =

y =

3. x = = 0

y =

y =

Question 3.

Find the ratio in which point T(-1, 6)divides the line segment joining the pointsP(-3, 10) and Q(6, -8).

Answer:

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

In the given question,

Let the point T divide the line PQ in the ratio m:n

Here x = -1 and y = 6

...(1)

.... (2)

Simplifying (1) we get,

-m-n = -3n + 6m

2n = 7m

Simplifying (2) we get,

6m + 6n = 10n-8m

14m = 4n

From both we get

Hence the point T divides PQ in the ratio 2:7

Question 4.

Point P is the centre of the circle and AB is a diameter. Find the coordinates ofpoint B if coordinates of point A and P are (2, -3) and (-2, 0) respectively.

Answer:

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:

In the given question A = (1,-3) and midpoint P is (-2,0).

Let coordinates of B be (c,d)

Then,

And

Solving for c and d, we get

-4 = 2 + c

c = -6

d = 3

Hence the coordinates of point B are (-6,3).

Question 5.

Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) andB(1, 2). Also find k.

Answer:

and

In the given question,

Let the point P divide AB is the ratio 1:k

Y coordinate of P

Simplifying

7k + 7 = 9k + 2

2k = 5

k =

and the ratio = 1:

=

Therefore point P divides AB in the ratio 2:5

Question 6.

Find the coordinates of midpoint of the segment joining the points (22,20) and (0, 16).

Answer:

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:

The coordinates of midpoint(x,y) are

Hence the coordinates are (11,18)

Question 7.

Find the centroids of the triangles whose vertices are given below.

(1) (-7, 6), (2, -2), (8, 5)

(2) (3, -5), (4, 3), (11, -4)

(3) (4, 7), (8, 4), (7, 11)

Answer:

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

1.

2.

3.

Question 8.

In ΔABC, G (-4, -7) is the centroid. If A (-14, -19) and B(3, 5) then find theco-ordinates of C.

Answer:

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

In the given question (x,y) = (-4,-7)

Hence

Solving for e, we get

e = -1

Solving for f, we get

f = -7

Hence the coordinates of the third point are (-1,-7)

Question 9.

A(h, -6), B(2, 3) and C(-6, k) are the co-ordinates of vertices of a trianglewhose centroid is G (1, 5). Find h and k.

Answer:

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

In the given question:

Solving for h we get

h = 7

Solving for k we get

k = 18

Question 10.

Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(-4, -8).

Answer:

let The points of trisection of a given line AB be P and Q

Then the ratio AP:PQ:QB = 1:1:1

Hence we get AP:PB = 1:2

And AQ:QB = 2:1

and

To find point P(x,y)

x = 0

y = 2

To find the point Q(x',y')

x' = -2

y' = -3

Hence point P = (0,2) and Q = (-2,-3)

Question 11.

If A (-14, -10), B(6, -2) is given, find the coordinates of the points whichdivide segment AB into four equal parts.

Answer:

let the points dividing AB be C,D,E.

AC:CD:DE:EB∷1:1:1:1

and

For C m:n ∷ 1:3

For D m:n ∷2:2

For E m:n ∷ 3:1

Hence coordinates of C = (-9,-8)

D = (-4,-6)

E = (1,-4)

Question 12.

If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts.

Answer:

Let the points dividing AB be C,D,E,F

AC:CD:DE:EF:FB∷1:1:1:1:1

and

For C m:n ∷ 1:4

For D m:n ∷ 2:3

For E m:n ∷ 3:2

= 8

For F m:n ∷ 4:1