Posted by anonymous on Tuesday, June 25, 2013 at 5:22am.
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  MathMatet, Tuesday, June 25, 2013 at 9:20am
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*69.5²89²
=>
x²=760
Substitute x² in (2):
4y²=89²760=7161
Finally, consider right triangle AKB:
AB²
=4x²+4y²
=3040+7161
=10201
=>
AB=101
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