Question

In the figure given below, PQR is an equilateral triangle of side 20 cm. ΔQSR is inscribed in it, ∠QSR = 90°, QS = 16 cm. Find (i) SR, (ii) the area of the shaded portion. [Take `sqrt3 = 1.732`].

Answer 1

Given:

• PQR is an equilateral triangle with each side = 20 cm
• ΔQSR is inscribed such that ∠QSR = 90°
• QS = 16 cm
• Need to find (i) SR and (ii) area of shaded portion
• Take `sqrt3 = 1.732`

Step 1: Calculate SR in ΔQSR

Since ∠QSR = 90°, triangle QSR is right angled at S.

Using Pythagoras theorem:

PR = 20 cm   ...(Side of equilateral triangle)

QR = 20 cm   ...(Equilateral triangle side)

Triangle PQR is equilateral, side = 20 cm.

In triangle QSR   ...(Right angled at S)

QS = 16 cm   ...(Given)

SR = ?

QR = 20 cm   ...(Base of equilateral triangle)

Using the Pythagoras theorem in ΔQSR:

(QS)² + (SR)² = (QR)²

⇒ (16)² + (SR)² = (20)²

⇒ 256 + (SR)² = 400

⇒ (SR)² = 400 – 256 = 144

⇒ SR = `sqrt(144)` = 12 cm

Step 2: Calculate area of the shaded portion

The shaded portion is the area of triangle PQR minus the area of triangle QSR.

Area of equilateral triangle PQR:

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

= `1.732/4 xx 20^2`

= `1.732/4 xx 400`

= 1.732 × 100

= 173.2 cm²

Area of triangle QSR right angled triangle:

`"Area" = 1/2 xx QS xx SR`

= `1/2 xx 16 xx 12`

= 96 cm²

i. SR = 12 cm

ii. Area of shaded portion 

= Area of triangle PQR − Area of triangle QSR 

= 173.2 − 96

= 77.2 cm²