Question

Find the area of the shaded region in the figure as shown, in which DPQS is an equilateral triangle and ∠PQR = 90°.

Answer 1

In right-angled ΔPPQR,
PR² = PQ² + RQ²
⇒ RQ² = PR² - PQ²
= 20² - 12²
= 400 - 144
= 256
⇒ RQ = 16cm
Area of ΔPQR

= `(1)/(2) xx "PQ" xx "RQ"`

= `(1)/(2) xx 12 xx 16`
= 96cm² 
Area of equilateral ΔPQS

= `sqrt(3)/(4) xx ("side")^2`

= `sqrt(3)/(4) xx 12 xx 12`
= `36sqrt(3)"cm"^2`
Now,
area of shaded region
= Area of ΔPQR - Area of equilateral ΔPQS
= 96 - `36sqrt(3)`
= `12(8 - 3sqrt(3))"cm"^2`.