How do you prove that "The altitude to the base of an isosceles triangle is also a median"?
1
Expert's answer
2012-03-20T11:25:12-0400
Let ABC be an isosceles triangle with base AC, so AB=BC. Let also BD be the altitude. We should prove that BD is also a median, so AD=DC.
Consider triangles ABD and CBD. Then we have the following:
1) Since ABC is isosceles, we have that angle A = angle C.
2) The angles BDA and BDC are right and so they are equal
3) It follows from 1) and 2) that the third angles ABD and CBD are equal as well.
3) AB=CB
5) the side BD is common for both triangles.
Thus due to 3), 4) and 5) thriangles ABD and CBD are equal, since they have two equal sides and equal angles between them. In particular, AD=CD, and so BD is median.
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