A circle touches the hypotenuse of a right-angled triangle at its midpoint and passes through the midpoint of the shorter side. If 3 units and 4 units are the length of the sides other than hypoteuse and r be the radius of the circle, then the value of 3 r is

Step-by-step solution

Using coordinate geometry, assume endpoints of hypotenuse as A ( 3 , 0 ) and B ( 0 , 4 ) , the third vertex as origin, such that the midpoint of the hypotenuse is P 3 2 , 2 and the midpoint of the shorter side is Q 3 2 , 0 Let centre be C ( h , k ) ∵ C P ⊥ A B ⇒ 2 − k 3 2 − h = 3 4 ⇒ 6 h − 8 k = − 7 … (i) ∵ C P = C Q ⇒ ( h − 3 2 ) 2 + ( k − 2 ) 2 = ( h − 3 2 ) 2 + k 2 ⇒ k = 1 and 6 h = 1 ⇒ h = 1 6 ∴ radius ( r ) = C Q = ( 1 6 − 3 2 ) 2 + 1 = ( 1 − 9 6 ) 2 + 1 = 5 3 ∴ 3 r = 5