Co­ordinate Geometry Exercise 14.1 (Part-9) | Extra Documents, Videos & Tests for Class 10 PDF Download

Question 51: If the coordinates of the mid-points of the sides of a triangle be (3, −2), (−3, 1) and (4, −3), then find the coordinates of its vertices.

Answer :

The co-ordinates of the midpoint  between two points  and  is given by,

Let the three vertices of the triangle be,  and.

The three midpoints are given. Let these points be,  and.

Let us now equate these points using the earlier mentioned formula,

Equating the individual components we get,

Using the midpoint of another side we have, 

Equating the individual components we get, 

Using the midpoint of the last side we have,

Equating the individual components we get, 

Adding up all the three equations which have variable ‘x’ alone we have, 

Substituting  in the above equation we have, 

Therefore, 

And 

Adding up all the three equations which have variable ‘y’ alone we have, 

Substituting  in the above equation we have, 

Therefore, 

And 

Therefore the co-ordinates of the three vertices of the triangle are.

Question 52: The line segment joining the points (3, −4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, −2) and (5/3, q) respectively. Find the values of p and q.

Answer : 

We have two points A (3,−4) and B (1, 2). There are two points P (p,−2) and Q  which trisect the line segment joining A and B.

Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,

The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2

Now we will use section formula to find the co-ordinates of unknown point A as,

Equate the individual terms on both the sides. We get, 

Similarly, the point Q is the point of trisection of the line segment AB. So, Q divides AB in the ratio 2: 1

Now we will use section formula to find the co-ordinates of unknown point A as,

Equate the individual terms on both the sides. We get,

Question 53:The line joining the points (2, 1) and (5, −8) is trisected at the points P and Q. If point P lies on the line 2x − y + k = 0. Find the value of k.

Answer :

We have two points A (2, 1) and B (5,−8). There are two points P and Q which trisect the line segment joining A and B.

Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,

The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2

Now we will use section formula to find the co-ordinates of unknown point A as,

Therefore, co-ordinates of point P is(3,−2)

It is given that point P lies on the line whose equation is

So point A will satisfy this equation.

So,

Question 54: A (4, 2), B(6, 5) and C (1, 4) are the vertices of ΔABC.
(i) The median from A meets BC in D. Find the coordinates of the point D.
(ii) Find the coordinates of point P and AD such that AP : PD = 2 : 1.
(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1
(iv) What do you observe?

Answer: 

We have triangle in which the co-ordinates of the vertices are A (4, 2); B (6, 5) and C (1, 4)

(i)It is given that median from vertex A meets BC at D. So, D is the mid-point of side BC.

In general to find the mid-point of two pointsand we use section formula as,

Therefore mid-point D of side BC can be written as, 

Now equate the individual terms to get, 

So co-ordinates of D is (ii)We have to find the co-ordinates of a point P which divides AD in the ratio 2: 1 internally.Now according to the section formula if any point P divides a line segment joining  and in the ratio m: n internally than,

P divides AD in the ratio 2: 1. So, 

(iii)We need to find the mid-point of sides AB and AC. Let the mid-points be F and E for the sides AB and AC respectively. 

Therefore mid-point F of side AB can be written as, 

So co-ordinates of F is 

 Similarly mid-point E of side AC can be written as,

So co-ordinates of E is 

Q divides BE in the ratio 2: 1. So, 

Similarly, R divides CF in the ratio 2: 1. So, 

(iv)We observe that that the point P, Q and R coincides with the centroid. This also shows that centroid divides the median in the ratio 2: 1. 

Question 55: If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

Answer : 

Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (6, 1); B (8, 2); C (9, 4) and D (k, p). 

Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide. 

In general to find the mid-point of two pointsand we use section formula as, 

The mid-point of the diagonals of the parallelogram will coincide. 

So,

Therefore, 

Now equate the individual terms to get the unknown value. So, 

Similarly, 

Therefore, 

Question 56: A point P divides the line segment joining the points A(3, −5) and B(−4, 8) such that If P lies on the line x + y = 0, then find the value of k.   

Answer : It is given that 

So, P divides the line segment joining the points A(3, −5) and B(−4, 8) in the ratio k : 1. Using the section formula, we get 

Coordinates of P = 

Since P lies on the line x + y = 0, so 

Hence, the value of k is 1/2 .

Question 57: The mid-point P of the line segment joining the points A(−10, 4) and B(−2, 0) lies on the line segment joining the points C(−9, −4) and D(−4, y). Find the ratio in which P divides CD. Also, find the value of y.      

Answer :

It is given that P is the mid-point of the line segment joining the points A(−10, 4) and B(−2, 0).  

∴ Coordinates of P =  ( -6 , 2)

Suppose P divides the line segment joining the points C(−9, −4) and D(−4, y) in the ratio k : 1. 

Using section formula, we get 

Coordinates of P =  

 ( -6 , 2)

Now, 

So, P divides the line segment CD in the ratio 3 : 2. 

Putting k 

⇒  3y - 8 =10

⇒ 3y=18 

⇒y=6 

Hence, the value of y is 6.