# math

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ABC is a triangle with AC=139 and BC=178. Points D and E are the midpoints of BC and AC respectively. Given that AD and BE are perpendicular to each other, what is the length of AB?

• math -

Since D, E bisect BC and AC respectively, we have
AE=139/2=69.5
BD=178/2=89

Let medians AD and BE meet at K.
Let KD=x, DE=y
Then
BK=2y, AK=2x
since medians intersect each other at third points.

Consider right triangle AKE right angled at K:
(2x)²+y²=69.5² (Pythagoras)
=>
4x²+y²=69.5²....(1)

Consider right triangle BKD,
x²+4y²=89².....(2)

4(1)-(2) to eliminate y:
15x²=4*69.5²-89²
=>
x²=760

Substitute x² in (2):
4y²=89²-760=7161

Finally, consider right triangle AKB:
AB²
=4x²+4y²
=3040+7161
=10201
=>
AB=101

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