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Geometry Level 5

A E AE is median on side B C BC and A B AB is angle bisector of ∠ D A E \angle DAE . If A E = 4 AE=4 units and B C = 10 BC=10 units. Then A B = a b AB=a\sqrt b

Find a + b a+b


The answer is 5.

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Aman Rajput by Aman Rajput , 25, India

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3 solutions

Let angle DAB be x = angle BAC . Angle ABC be y . Now angle CAE IS 180-2x . Using sine rule in triangle ABE sin(y) = 0.8* sin(x) . Now using sine rule in triangle ABC AC= 10*sin(y) / sin(x) = 8 . Now by appolinius theoram AB^ 2 + AC^2= 2(AE^2 + BE^2) we get AB as thrice of square root of two . Hence answer = 3+2=5

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Sauvik Mondal
Oct 9, 2015

Let the parallel from E to AB cuts AC at F.So F will be midpoint of AC.Now <DAB=<BAE=<AEF=<AFE(EF||AB).So AE=AF=4.Now using appolonius theorem on ABC we get AB^2+AC^2=2(BE^2+AE^2) and hence a+b=5.

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Nihar Mahajan
Sep 3, 2015

<No solution>

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