Question

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

(a) 3 cm 

(b) 2 cm 

(c) 5 cm 

(d) 4 cm

Answer 1

(b) 2 cm

Let ΔABC be right angled at A. 

Since the incentre is equidistant from the sides, let the radius of the incircle be r.

∴ OP = OQ = OR = r cm

AC² + AB² = BC² 

⇒ BC² = 6² + 8² = 36 + 64 =100 

⇒ BC = 10 cm. Now, 

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

⇒ \(\frac12\times{AB}\times{AC}=\frac12\times{r}\times{AB}\) + \(\frac12\times{r}\times{BC}\) + \(\frac12\times{r}\times{CA}\)

⇒ \(\frac12\times{6}\times{8}\) = \(\frac12\times{r}\times{6}\) + \(\frac12\times{r}\times{10}\) + \(\frac12\times{r}\times{8}\)

⇒ 12 r = 24 ⇒ r = 2 cm.