Question

ABC is a right-angled triangle right angled at A. A circle is inscribed in it the lengths of the two sides containing the right angle are 6 cm and 8 cm. Find the radius of the circle.

Answer 1

Using Pythagoras theorem in ΔBAC, we have

$\Rightarrow$${BC}^2$ = ${AB}^2$ + ${AC}^2$

$\Rightarrow$${BC}^2$ = $6^2$ + $8^2$ = 100 ⇒ BC = 10 cm

Now, Area of ΔABC = Area of ΔOAB + Area of ΔOBC + Area of ΔOCA

$\Rightarrow$ $\frac{1}{2}$AB $\times$ AC = $\frac{1}{2}$AB $\times$ r + $\frac{1}{2}$BC $\times$ r + $\frac{1}{2}$CA $\times$ r

$\Rightarrow$ $\frac{1}{2}$6 $\times$ 8 = $\frac{1}{2}$(6 $\times$ r) + $\frac{1}{2}$(10 $\times$ r) + $\frac{1}{2}$(8 $\times$ r)

$\Rightarrow$ 48 = 24r

$\Rightarrow$ r = 2 cm