Q. 284.2( 12 Votes )

# In the given figure, ABC is an equilateral triangle, PQ ‖AC and AC is produced to R such that CR=BP. Prove that QR bisects PC.

Answer :

Given: ABC is an equilateral triangle, PQ ‖AC and CR=BP

To prove: QR bisects PC or PM = MC

Proof:

Since, ∆ABC is equilateral triangle,

∠A = ∠ACB = 60°

Since, PQ ‖AC and corresponding angles are equal,

∠BPQ = ∠ACB = 60°

In ∆BPQ,

∠B= ∠ACB = 60°

∠BPQ = 60°

Hence, ∆BPQ is an equilateral triangle.

∴ PQ = BP = BQ

Since we have BP = CR,

We say that PQ = CR …(1)

Consider the triangles ∆PMQ and ∆CMR,

∠PQM = ∠CRM …alternate angles

∠PMQ = ∠CMR … vertically opposite angles

PQ = CR … from 1

Thus by AAS property of congruence,

∆PMQ ≅ ∆CMR

Hence, we know that, corresponding parts of the congruent triangles are equal

∴ PM = MC

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