A geometry problem by Hana Wehbi

Geometry Level pending

In the diagram above, D A = A B = B E DA=AB=BE , G A = A C = C F GA=AC=CF and I C = C B = B H IC= CB=BH .

If E F = 5 EF= 5 , D I = 6 DI=6 , and G H = 7 GH=7 , what is the area of △ A B C \triangle ABC ?

Problem and image: courtesy Lehigh University

5 6 2 \frac{5\sqrt{6}}{2} 3 2 2 \frac{3\sqrt{2}}{2} 3 6 2 \frac{3\sqrt{6}}{2} 2 3 4 \frac{2\sqrt{3}}{4} 7 6 2 \frac{7\sqrt{6}}{2}

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Mar 7, 2017

By similar triangles, A C = 1 2 D I AC=\frac{1}{2}DI , A B = 1 2 G H AB = \frac{1}{2}GH and C B = 1 2 F E CB=\frac{1}{2}FE .

Thus the area of a △ A B C \triangle ABC is 1 4 \frac{1}{4} times the area of a triangle with sides 5 , 6 , 7 5,6,7 , which, by Heron's formula is 9.2.3.4 \sqrt{9.2.3.4} = 1 4 × 9.2.3.4 = 3 6 2 \frac{1}{4} \times \sqrt{9.2.3.4}= \frac {3\sqrt{6}}{2}

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