Question

Figure shows an equilateral triangle ABC whose each side is 10 cm and a right-angled triangle BDC whose side BD = 8 cm and ∠D = 90°. Find the area of the shaded portion.

Answer 1

Given:

ABC is an equilateral triangle with side AB = BC = CA = 10 cm.

Triangle BDC is right-angled at D with BD = 8 cm and BC = 10 cm.

Step-wise calculation:

1. Use Pythagoras in right triangle BDC to find DC:

`DC = sqrt(BC^2 - BD^2)`

= `sqrt(10^2 - 8^2)`

= `sqrt(100 - 64)`

= `sqrt(36)`

= 6 cm

2. Area of equilateral triangle ABC (side a = 10):

Area (ABC) = `sqrt(3)/4 a^2` 

= `sqrt(3)/4 xx 10^2` 

= `25sqrt(3)  cm^2`

3. Area of right triangle BDC (legs 6 and 8):

Area (BDC) = `1/2` × BD × DC 

= `1/2 xx 8 xx 6`

= 24 cm²

4. Shaded area = Area (ABC) – Area (BDC) 

= `25sqrt(3) - 24  cm^2` 

= 25(1.73205) – 24 

= 43.3013 – 24 

= 19.3013 cm²

Shaded area = `25sqrt(3)` – 24 cm² ≈ 19.30 cm².