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Question

In a triangle ABC, AB=AC. Show that the altitude AD is median also.

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Solution

Given: ABC is an isosceles triangle(AB=AC)
AD is the altitude
To prove: AD is the median
Proof:
AB=AC (given)
AD is common
∠ADB=∠ADC=90° (AD is the height)
∴By RHS criteria, ΔABD is congruent to ΔACD

By CPCT,
BD=CD
And area of congruent triangles are equal.

A median from a vertex divides the opposite side into two halves and the two triangles formed are equal in area.

Hence AD is the median as well as it's height


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