Question

In ΔPQR, ∠Q = 90°. QM is the median of the triangle through Q. Prove that QM = `1/2` PR.

Answer 1

Step 1:

Since ΔPQR is a right-angled triangle with ∠Q = 90°, its circumcenter lies at the midpoint of the hypotenuse PR.

M is the midpoint of PR because QM is a median to PR.

Therefore, M is the circumcenter of ΔPQR.

Step 2:

The distance from the circumcenter to each vertex of the triangle is equal to the circumradius.

So, QM = PM = MR.

Step 3:

Since M is the midpoint of PR, PR = PM + MR.

As PM = MR, we have PR = PM + PM = 2PM.

Similarly, PR = 2MR.

Step 4:

From Step 2, QM = PM.

From Step 3, PM = `1/2` PR.

Therefore, QM = `1/2` PR.

The length of the median QM from the right angle to the hypotenuse PR is half the length of the hypotenuse PR.