Question 826371: Given: AD is the median to BC; AD is the altitude to BC
Prove: Triangle ABC is isosceles
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! If AD is the median to BC, D is the midpoint of BC.
If D is the midpoint of BC, BD=DC.
If AD is the altitude to BC, angles BDA and CDA are right angles.
So right triangles ABD and ACD have
congruent legs AD and BC,
congruent legs AD and AD,
and congruent angles ADb and ADC.
The two right triangles are congruent by SAS congruency
(two sides and the angle between them from one triangle, are congruent to corresponding parts in the other triangle.
Since Congruent Parts in Congruentr Triangles are Congruent (CPCTC),
hypotenuses AB and AC are congruent.
Since AB and AC are congruent, ABC is an isosceles triangle.
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