Question

Triangle PQR is inscribed in a semicircle. PQ = PR = 7 cm. Find the shaded area.

Answer 1

Given:

• Triangle PQR is inscribed in a semicircle.
• PQ = PR = 7 cm.
• ∠PQR = 90° (angle subtended in a semicircle is right angle).

We need to find the shaded area (area of triangle PQR).

Step-wise solution:

Since PQ = PR = 7 cm, triangle PQR is isosceles and right-angled at Q.

We know PQR is inscribed in a semicircle, so QR is the diameter of the semicircle.

To find QR, use the Pythagoras theorem in triangle PQR: 

`QR^2 = PQ^2 + PR^2 = 7^2 + 7^2 = 49 + 49 = 98`

`QR = sqrt98 = 7sqrt2` cm

The radius of the semicircle (r) is half of QR:

`r = (QR)/2 = (7sqrt2)/2` cm

Area of the triangle PQR: Since PQ and PR are equal and triangle is right-angled at Q,

`"Area" = 1/2 xx PQ xx PR`

= `1/2 xx 7 xx 7`

`48/2 = 24.5  cm^2`

Area of the semicircle:

`1/2pir^2 = 1/2 xx 22/7 xx ((7sqrt2)/2)^2`

`1/2 xx 22/7 xx (49 xx 2)/4`

`1/2 = 22/7 xx 98/4`

`1/2 xx 22/7 xx 24.5`

11 × 3.5 = 38.5 cm²

Shaded area is the area of the semicircle minus the area of the triangle:

Shaded area = 38.5 − 24.5 = 14 cm²