Question

Prove that if altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is isosceles.

Answer 1

Given:

In triangle △PQR,

• PN ⊥ QR
• PM ⊥ QR
• PN = PM   ...(The altitudes from vertices Q and R are equal)

To Prove:

PQ = PR → triangle △PQR is isosceles.

Proof:

Consider triangles △PNQ and △PMR:

In △PNQ and △PMR:

1. PN = PM   ...(Given, altitudes are equal)
2. ∠PNQ = ∠PMR = 90°   ...(Each is a right angle)
3. ∠QPN = ∠RPM   ...(Same angle at vertex P)

So, by RHS (Right angle–Hypotenuse–Side) congruence rule △PNQ ≅ △PMR.

Therefore, PQ = PR   ...(Corresponding parts of congruent triangles)

Thus, △PQR is isosceles.