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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.
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Solution
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
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