Question

Triangle ABC is inscribed in a semicircle. AB = 12 cm, BC = 16 cm. Find

1. AC 
2. the shaded area. [Take π = 3.14]

Answer 1

Given:

• Triangle ABC is inscribed in a semicircle
• AB = 12 cm
• BC = 16 cm
• π = 3.14

We need to find:

1. AC 
2. The shaded area

Step-wise calculation:

i. Find AC using the Pythagorean theorem, since triangle ABC inscribed in a semicircle implies ∠B = 90° (right angle):

`AC = sqrt(AB^2 + BC^2)`

= `sqrt(12^2 + 16^2)`

= `sqrt(144 + 256)`

= `sqrt(400)`

= 20 cm

ii. Find the area of triangle ABC:

`"Area of"  ΔABC = 1/2 xx AB xx BC`

= `1/2 xx 12 xx 16`

= 96 cm² 

iii. Find the radius (r) of the semicircle:

Since AC is the diameter,

`r = (AC)/2`

= `20/2`

= 10 cm

iv. Find the area of the semicircle:

Area of semicircle = `1/2 πr^2`

= `1/2 xx 3.14 xx 10^2`

= 157 cm²

v. Find the shaded area:

The shaded area is the area of the semicircle minus the area of triangle ABC.

Shaded area = 157 – 96

= 61 cm²

Thus, AC = 20 cm and shaded area = 61 cm²