Answer the Following:

Question

Given: Triangle ABC with median line AD is perpendicular to line BC

Prove: Line AD bisects angle BAC and triangle ABC is isosceles

Answer 1

The median line AD splits BC into two equal lengths of BC/2.

Since AD ⊥ BC ⇒ AD splits ΔABC into two right triangles ΔABD and ΔACD.

Both triangles have sides of length AD and BC/2 ⇒ AB = AC.

Thus ΔABD ≅ ΔACD (congruent) ⇒ ∠BAD = ∠CAD = ∠BAC/2.

Since ΔABC has two equal sides AB = AC ⇒ isosceles.