Show that the triangle whose vertices are A ( 8 , - 4 ) , B ( 9 , 5 ) and C ( 0 , 4 ) is an isosceles triangle.

Given, A ( 8 , - 4 ) , B ( 9 , 5 ) and C ( 0 , 4 ) . We know that, distance between points A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is given by: A B = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 . ∴ A B = 9 - 8 2 + 5 - ( - 4 ) 2 = 1 2 + 9 2 = 1 + 81 = 82 units B C = 0 - 9 2 + 4 - 5 2 = 9 2 + - 1 2 = 81 + 1 = 82 units A C = 0 - 8 2 + 4 - - 4 2 = 8 2 + 8 2 = 128 = 8 2 units Here, sides A B and B C are equal in △ A B C . Therefore, the △ A B C is an isosceles triangle.

Find the co-ordinates of the mid-point of the line segment joining the points 2 , 3 and 4 , 7 .

Let the given points be : A ( 2 , 3 ) and B ( 4 , 7 ) We know that, the Mid-point of ( a , b ) and ( c , d ) is a + c 2 , b + d 2 Now, the Mid-point of A ( 2 , 3 ) and B ( 4 , 7 ) = 2 + 4 2 , 3 + 7 2 = ( 3 , 5 ) Hence, the mid-point of pair of points ( 2 , 3 ) and ( 4 , 7 ) is ( 3 , 5 ) .

Find the coordinates of the point which divides the line segment joining the point A 4 , - 3 & B 8 , 5 in ratio 3 : 1 internally.

The point that divides the points x 1 , y 1 and x 2 , y 2 in the ratio m : n is m x 2 + n x 1 m + n , m y 2 + n y 1 m + n Let C divides the line segment joining the point A ( 4 , - 3 ) and B ( 8 , 5 ) in the ratio 3 : 1 internally. C = 3 ( 8 ) + 1 ( 4 ) 3 + 1 , 3 ( 5 ) + 1 ( - 3 ) 3 + 1 = 24 + 4 4 , 15 - 3 4 = 28 4 , 12 4 = ( 7 , 3 ) Therefore, the required point is ( 7 , 3 ) .

Find the coordinates of a point A , where AB is a diameter of a circle whose centre is 2 , - 3 and B is 1 , 4 .

We know that the centre is the mid-point of the two extreme points of the diameter in a circle. So, here, O is the mid-point of the diameter A B . Let the coordinate of A be h , k . So, we have, h + 1 2 = 2 [The coordinates of the mid-point of the line segment joining the points x 1 , y 1 and x 2 , y 2 is x 1 + x 2 2 , y 1 + y 2 2 ] ⇒ h + 1 = 4 ⇒ h = 4 - 1 ⇒ h = 3 Also, we get, k + 4 2 = - 3 ⇒ k + 4 = - 6 ⇒ k = - 6 - 4 ⇒ k = - 10 Therefore, the coordinate of the point A is 3 , - 10 .