GRE problem

Geometry Level 3

If A B = B C AB=BC , A B = a AB=a , D E = b DE=b , and ∠ A B C = ∠ E D C = 9 0 ∘ \angle ABC =\angle EDC = 90^{\circ} then which of the following is an expression for the area of quadrilateral A B D E ? ABDE?

A. a 2 2 − b 2 2 \large\frac{a^2}{2}-\frac{b^2}{2}

B. a 2 2 + b 2 2 \large\frac{a^2}{2}+\frac{b^2}{2}

C. a 2 4 − a b 2 \large\frac{a^2}{4}-\frac{ab}{2}

D. a 2 − b 2 \large a^2-b^2

E. a 2 4 + a b 2 \large\frac{a^2}{4}+\frac{ab}{2}

E B C D A

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Marta Reece
Jul 3, 2017

[ A B C ] = a 2 2 [ABC]=\frac{a^2}2 Both legs of the triangle are given and equal a \small{\color{#3D99F6}\text{ Both legs of the triangle are given and equal }a}

[ C D E ] = b 2 2 [CDE]=\frac{b^2}2 Both legs have to again be the same and equal b \small{\color{#3D99F6}\text{ Both legs have to again be the same and equal }b}

[ A B D E ] = [ A B C ] − [ C D E ] = a 2 2 − b 2 2 [ABDE]=[ABC]-[CDE]=\boxed{\frac{a^2}2-\frac{b^2}2}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you.

Hana Wehbi - 3 years, 11 months ago

An other way to find out the answer, is to look at a special case: DE is a midline of ABC. Then a=2b and [CDE]=1/4[ABC]. By a little calculating in your mind you can easily get the answer.

Áron Bán-Szabó - 3 years, 11 months ago

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That's only multiple guessing from existing answers rather than actually solving the problem, however.

Marta Reece - 3 years, 11 months ago

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I know, but I had a lot of competitions like this, And there this is really helpfully,

Aa Ás - 3 years, 11 months ago

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