Question

ΔABC is an isosceles triangle in which AB = AC. If P, Q and R are the mid-points of the sides AB, BC and CA, respectively, prove that ΔPQR is also an isosceles triangle.

Answer 1

Given: In ΔABC, AB = AC. P, Q, R are mid-points of AB, BC, CA respectively.

To Prove: ΔPQR is isosceles i.e., two of its sides are equal.

Proof [Step-wise]:

1. Since P and R are mid‑points of AB and AC. 

`PB = 1/2 xx AB` and `RC = 1/2 xx AC`.

Because AB = AC, PB = RC.

2. Since Q is the mid‑point of BC, BQ = QC.

3. In ΔABC,

AB = AC 

⇒ ∠ABC = ∠ACB   ...(Base angles of an isosceles triangle are equal).

4. Consider triangles PBQ and QCR:

PB = RC   ...(From step 1),

BQ = QC   ...(From step 2),

∠PBQ = ∠RCQ because ∠PBQ = ∠ABC and ∠RCQ = ∠ACB and those are equal.   ...(Step 3) 

Hence, by SAS, ΔPBQ ≅ ΔQCR.

5. From the congruence,

Corresponding sides PQ and RQ are equal.

Therefore, PQ = RQ.

So, ΔPQR has two equal sides.

As an alternative remark: the segment joining mid‑points of AB and AC is parallel to BC by the midpoint theorem, which is a standard result about mid‑point segments. 

ΔPQR is isosceles PQ = RQ.