Question

In a rectangle, the angle between a diagonal and a side is 30° and the length of this diagonal is 8 cm. The area of the rectangle is ______.

Options

• 16 cm²

• `16/sqrt(3)` cm²

• `16sqrt(3)` cm²

• `8sqrt(3)` cm²

Answer 1

In a rectangle, the angle between a diagonal and a side is 30° and the length of this diagonal is 8 cm. The area of the rectangle is `underlinebb(16sqrt(3)  cm^2)`.

Explanation:

Let ABCD be the rectangle.

`(BC)/(AC) = sin 30^circ = 1/2`

⇒ `x/(8  cm) = 1/2`

⇒ x = 4 cm

∴ AB² = (8)² cm² – (4)² cm²

= (64 – 16) cm²

= 48 cm²

⇒ `AB = sqrt(48  cm^2)`

⇒ `AB = 4sqrt(3)  cm`

∴ `ar(ABCD) = (4sqrt(3) xx 4) cm^2`

= `16sqrt(3)  cm^2`