Question

In the adjoining figure, ABC is an equilateral triangle in which BC = 10 cm and BCD is a right-angled triangle in which BD = CD. Find the area of the shaded region.

Answer 1

Given:

ABC is equilateral with BC = 10 cm.

Triangle BCD is right-angled at D with BD = CD so BDC is an isosceles right triangle.

Step-wise calculation:

1. In an isosceles right triangle the hypotenuse = `"leg"·sqrt(2)`.

Here, BC is the hypotenuse. 

So, each leg BD = CD

= `"BC"/sqrt(2)`

= `10/sqrt(2)`

= `5sqrt(2)` cm

2. Area of ΔBCD (right triangle with legs BD and CD):

Area BCD = `1/2` × BD × CD 

= `1/2 xx (5sqrt(2)) xx (5sqrt(2))` 

= `1/2 xx 50` 

= 25 cm²

3. Area of equilateral ΔABC with side a = 10 is

Area ABC = `sqrt(3)/4 xx a^2` 

= `sqrt(3)/4 xx 100` 

= `25sqrt(3)` cm²

4. Shaded area = Area ABC − Area BCD 

= `25sqrt(3) - 25`

= `25(sqrt(3) - 1)` cm²

Shaded area = `25(sqrt(3) - 1)` cm²

= 25(1.732 – 1)

= 18.3 cm²