Question

In the rectangle PQRS, ∠SQR = 60° and QR = 4 cm. Find SR and QS.

Answer 1

Given:

□PQRS is a rectangle

∠SQR = 60°

QR = 4 cm

Need to find SR and QS

Step 1:

In triangle SQR:

Right-angled triangle at R

QR is the vertical side

SR is the base (adjacent to ∠SQR)

QS is the hypotenuse

Step 2: Trigonometry

Use tan(θ) to find SR:

tan(60°) = `("opposit"(QR))/("adjacent"(SR))`

= `sqrt3 = 4/(SR)`

= SR = `4/sqrt3`

= `(4sqrt3)/3` cm

cot(60°) = `(SR)/(QR)`

= `1/sqrt3 = (SR)/4`

SR = `4sqrt3` cm

∴ cot(60°) = `(SR)/(QR) = 1/sqrt3`

`(SR)/4 = SR`

= `4sqrt3`cm

Step 3:

= QS² = QR² + SR²

= `4^2 + (4sqrt3)^2`

= 16 + 48 = 64

QS = `sqrt64`

= 8 cm