Question

In a rectangle ABCD, T is a point on AD such that ∆BTC is equilateral. If BC = 4 cm, find the area of ABCD.

Answer 1

Given:

ABCD is a rectangle.

T lies on AD such that triangle △BTC is equilateral.

BC = 4 cm

Find the area of rectangle ABCD.

Step 1: Use property of equilateral triangle

In △BTC, since it’s equilateral, all sides are equal:

BT = TC = BC = 4 cm

Step 2: Drop perpendicular from T to BC

Let’s find the height of equilateral triangle BTC. 

Height h of an equilateral triangle of side a is:

`h = sqrt3/2 ⋅ a = sqrt3/2 ⋅ 4 = 2sqrt3` cm

Step 3:

Since BC = 4 cm 

Height AB = height of rectangle = h = `2sqrt3` cm

Area = AB × BC = `2sqrt3 xx 4`

= `8sqrt3` cm²