Question

A circle is inscribed in ΔABC, AB = AC = 15 cm, BC = 24 cm. Calculate

1. the radius of incircle 
2. the area of the shaded region. [Take π = 3.14]

Answer 1

Given:

• Triangle ABC with sides AB = AC = 15 cm (isosceles), BC = 24 cm.
• A circle is inscribed in the triangle.
• π = 3.14.
• Need to find (i) radius of incircle, (ii) area of shaded region (triangle area minus incircle area).

Step-wise calculation:

1. Calculate semi-perimeter (s) of triangle ABC:

`s = (AB + BC + AC)/2`

= `(15 + 24 + 15)/2`

= `54/2`

= 27 cm

2. Calculate the area (Δ) of triangle ABC using Heron’s formula:

`Δ = sqrt(s(s - AB)(s - BC)(s -AC))`

= `sqrt(27(27 - 15)(27 - 24)(27 - 15))`

Calculate each term:

(27 – 15) = 12,

(27 – 24) = 3,

(27 – 15) = 12

Thus,

`Δ = sqrt(27 xx 12 xx 3 xx 12)`

= `sqrt(27 xx 432)`

= `sqrt(11664)`

`sqrt(11664) = 108  cm^2`

3. Calculate radius (r) of the incircle by formula:

`r = ("Area of"  Δ)/"Semi-perimeter"`

= `108/27`

= 4 cm

4. Calculate area of the incircle:

`"Area"_("incircle") = πr^2`

= 3.14 × 4²

= 3.14 × 16

= 50.24 cm²

5. Calculate area of shaded region (area of triangle minus area of incircle):

Shaded area = 108 – 50.24 = 57.76 cm²

Thus, radius of the incircle r = 4 cm and area of the shaded region = 57.76 cm².