Question

PQR is an equilateral triangle. QM and RN are medians. Prove that QM = RN.

Answer 1

Given: The triangle △PQR is equilateral and QM and RN are medians.

To Prove:

QM = RN

Proof:

Since △PQR is equilateral:

PQ = QR = RP

All angles are 60°.

Medians from all vertices in an equilateral triangle are equal in length.

In an equilateral triangle:

All sides and angles are equal.

The medians are also angle bisectors, altitudes and perpendicular bisectors.

Hence, all medians are equal in length.

Therefore, QM = RN and also equal to the third median from P.

Hence proved QM = RN in an equilateral triangle.