Practice Set 5.2 Co-ordinate Geometry Class 10th Mathematics Part 2 MHB Solution

Practice Set 5.2

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

Practice Set 5.2
Question 1.

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:

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 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


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 


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


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 


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


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 


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



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