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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