Co­ordinate Geometry Exercise 14.1 (Part-6) | Extra Documents, Videos & Tests for Class 10 PDF Download

Question 7: Find the ratio in which the point (2, y) divides the line segment joining the points A (−2, 2) and B (3, 7). Also, find the value of y.

Answer :

The co-ordinates of a point which divided two points  and  internally in the ratio is given by the formula,

Here we are given that the point P(2,y) divides the line joining the points A(−2,2) and B(3,7) in some ratio.

Let us substitute these values in the earlier mentioned formula.

Equating the individual components we have

We see that the ratio in which the given point divides the line segment is.

Let us now use this ratio to find out the value of ‘y’.

Equating the individual components we have

y = 6

Thus the value of ‘y’ is 6 .

Question 8: If A (−1, 3), B (1, −1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Answer :

The distance d between two points  and  is given by the formula

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

Here, it is given that the three vertices of a triangle are A(−1,3), B(1,−1) and C(5,1).

The median of a triangle is the line joining a vertex of a triangle to the mid-point of the side opposite this vertex.

Let ‘D’ be the mid-point of the side ‘BC’.

Let us now find its co-ordinates.

Thus we have the co-ordinates of the point as D(3,0).

Now, let us find the length of the median ‘AD’.

Thus the length of the median through the vertex ‘A’ of the given triangle is.

Question 9: If the points P, Q(x, 7), R, S(6, y) in this order divide the line segment joining A(2, p) and B(7, 10) in 5 equal parts, find x, y and p. 

Answer :

It is given that P, Q(x, 7), R, S(6, y) divides the line segment joining A(2, p) and B(7, 10) in 5 equal parts.
∴ AP = PQ = QR = RS = SB          .....(1) 

Now,
AP + PQ + QR + RS + SB = AB 

⇒ SB + SB + SB + SB + SB = AB             [From (1)]
⇒ 5SB = AB 

⇒ SB = 1/5 AB                  .....(2)    

Now,
AS = AP + PQ + QR + RS =   

From (2) and (3), we get 

Similarly,
AQ : QB = 2 : 3
Using section formula, we get 

Coordinates of Q =   

Now, 

Coordinates of S =  

(6,y)=(6,9) 

⇒y=9 

Thus, the values of x, y and p are 4, 9 and 5, respectively. 

Question 10: If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (−2,−3) and (5 2) find the other vertices.

Answer :

Let a  in which P and Q are the mid-points of sides AB and AC respectively. The coordinates are: A (1, 1); P (−2, 3) and Q (5, 2).

We have to find the co-ordinates of and.

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

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

Now equate the individual terms to get,

So, co-ordinates of B is (−5, 5)

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

 Now equate the individual terms to get,

So, co-ordinates of C is (9, 3)

Let A (−4,−1); B (−2,−4); C (4, 0) and D (2, 3) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rectangle.

So we should find the lengths of opposite sides of quadrilateral ABCD.

Opposite sides are equal. So now we will check the lengths of the diagonals.

Opposite sides are equal as well as the diagonals are equal. Hence ABCD is a rectangle.

Question 18: Find the lengths of the medians of a triangle whose vertices are A (−1,3), B(1,−1) and C(5, 1).

Answer : 

We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (−1, 3); B (1,−1) and C (5, 1).

So we should find the mid-points of the sides of the triangle.

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

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

Now equate the individual terms to get,

So co-ordinates of P is (0, 1)

Similarly mid-point Q of side BC can be written as,

Now equate the individual terms to get,

So co-ordinates of Q is (3, 0)

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

Now equate the individual terms to get,

So co-ordinates of Q is (2, 2)

Therefore length of median from A to the side BC is,

Similarly length of median from B to the side AC is,

Similarly length of median from C to the side AB is

Question 19: Find the ratio in which the line segment joining the points A(3, −3) and B(−2, 7) is divided by the x-axis. Also, find the coordinates of the point of division.   

AnswerSuppose the x-axis divides the line segment joining the points A(3, −3) and B(−2, 7) in the ratio k : 1.
Using section formula, we get
Coordinates of the point of division = 

 Since the point of division lies on the x-axis, so its y-coordinate is 0.

So, the required ratio is 3/7 : 1 or 3 : 7. 

Putting k = 3/7, we get 

Coordinates of the point of division =  

Thus, the coordinates of the point of division are 

Question 20: Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of x.      

Answer :Suppose P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3) in the ratio k : 1.
Using section formula, we get
Coordinates of P = 

Now, 

So, P divides the line segment AB in the ratio 3 : 5. 

Thus, the value of x is 9.