Question

ABC is a right triangle, right angled at B. 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 incircle. [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 3 Marks

Let the radius of the incircle be x cm.

Let the incircle touch the side AB, BC and CA at D, E, F respectively.

Let O be the centre of the circle.

Then, OD = OE = OF = x cm.

Also, AB = 8 cm and BC = 6 cm.

Since the tangents to a circle from an external point are equal, we have

AF = AD = (8 – x) cm, and

CF = CE = (6 – x) cm.

Therefore, AC = AF + CF = (8 – x) cm + (6 – x) cm

= (14 – 2x) cm.

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

$\Rightarrow {(14-2x)}^2=8^2+6^2=100={\left(10\right)}^2$

$\Rightarrow 14–2x=\pm 10\Rightarrow x=2orx=12$

$\Rightarrow x=2$ [neglecting x = 12].

Hence, the radius of the incircle is 2cm.