Question

In following fig., ABC is a right- angled triangle at A with sides AB = 5 cm and BC = 13 cm . A circle with centre O has been inscribed in the triangle ABC. Calculate the radius of the incircle. 

Answer 1

In right Δ BAC, 

BC²= AC²+ AB² 

AC²= 13² -5² 

AC²  = 169 - 25

AC²  = 144

AC = 12

Let OP = OQ = r (say) (radius of same circle) 

∠ OQP = ∠ OPQ = 90° (radius is .L to tangent at the point of contact) 

∴ OPAQ is a square. 

AQ = AP = OP = OQ = r 

BQ = BR = 5 - r ---(1)    (length of tangents drawn from an external point) 

PC = CR = 12 - r - (2)  to a circle are equal} 

BC = CR+ BR 

13 = 12 - r + 5 - r    [from (1) and {2}] 

2r = 4

r = 2

Thus , radius of the circle is 2 cm