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The quadrilateral ABCD is a kite. If AB = BC = 20, CD = DA =

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The quadrilateral ABCD is a kite. If AB = BC = 20, CD = DA =

by Max@Math Revolution » Mon Jun 10, 2019 11:23 pm

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[GMAT math practice question]

The quadrilateral ABCD is a kite. If AB = BC = 20, CD = DA = 15 and BD = 7, what is the area of the kite ABCD shown in the figure below?

Image

A. 60
B. 72
C. 84
D. 96
E. 108
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by Max@Math Revolution » Wed Jun 12, 2019 6:11 pm
=>

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Let x and y be the lengths of DE and CE, repectively.
From the right triangle CDE, we have CD^2 = x^2 + y^2 or x^2+y^2=225.
From the right triangle BCD, we have BC^2 = (x+7)^2 + y^2 or (x+7)^2 + y^2 = 400.
Subtracting the first equation from the second yields (x+7)^2 - x^2 = (x+7+x)(x+7-x) = 7(2x+7) = 175, and x = 9.
y = 12 since y^2 = 225 - x^2 = 225 - 81 = 144.
The area of the kite ABCD is the area of triangle ABC - the area of triangle ADC = (1/2)(2y)(x + 7) - (1/2)(2y)x = (1/2)(24)(16) - (1/2)(24)(9) = (1/2)(24)(16-9) = 12*7 = 84.

Therefore, the answer is C.
Answer: C
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