Question

Calculate the area of quadrilateral ABCD, in which ∠ABD = 90°, triangle BCD is an equilateral triangle of side 24 cm and AD = 26 cm.

Answer 1

Consider the figure: 

From the right triangle ABD we have 

AB = `sqrt(26^2 - 24^2)`

= `2sqrt(13^2 - 12^2)`

= 2 × 5

= 10

The area of right triangle ABC will be

ΔABD = `1/2("AB")("BD")`

= `1/2``(10) (24)`

= 120

Again from the equilateral triangle BCD, we have CP ⊥ BD

PC = `sqrt(24^2 - 12^2)`

= `12sqrt(2^2 - 1^2)`

= `12sqrt(3)`

Therefore, the area of the triangle BCD will be

ΔBCD = `1/2 ("BD")( "PC" )`

= `1/2 (24) (12sqrt3)`

= `144sqrt3`

Hence, the area of the quadrilateral will be

ΔABD + ΔBCD

= 120 + `144sqrt3`

= 369.41 cm²