In the given figure, ABC is an equilateral triangle; PQ || AC and AC i

1.	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» It is given that ABC is an equilateral triangle; PQ \|\| AC and AC is produced to R such that CR = BP Consider QR intersecting the line PC at point M We know that △ ABC is an equilateral triangle So we get ∠ A = ∠ ACB = 60^o From the figure we know that PQ \|\| AC and ∠ BPQ and ∠ ACB are corresponding angles So we get ∠ BPQ = ∠ ACB = 60^o Based on the △ BPQ we know that ∠ B = ∠ ACB = 60^o It can be written as ∠ BQP = 60^o According to the figure we know that △ BPQ is an equilateral triangle So we get PQ = BP = BQ It is given that CR = BP so we get PQ = CR ……. (1) In the △ PMQ and △ CMR we know that PQ \|\| AC and QR is the transversal We know that ∠ PQM and ∠ CRM are alternate angles and ∠ PMQ and ∠ CMR are vertically opposite angles ∠ PQM = ∠ CRM ∠ PMQ = ∠ CMR By considering equation (1) and AAS congruence criterion △ PMQ ≅ △ CMR We know that the corresponding parts of congruent triangles are equal PM = MC Therefore, it is proved that QR bisects PC.

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»

It is given that ABC is an equilateral triangle; PQ || AC and AC is produced to R such that CR = BP

Consider QR intersecting the line PC at point M

We know that △ ABC is an equilateral triangle

So we get ∠ A = ∠ ACB = 60^o

From the figure we know that PQ || AC and ∠ BPQ and ∠ ACB are corresponding angles

So we get

∠ BPQ = ∠ ACB = 60^o

Based on the △ BPQ we know that

∠ B = ∠ ACB = 60^o

It can be written as

∠ BQP = 60^o

According to the figure we know that △ BPQ is an equilateral triangle

So we get

PQ = BP = BQ

It is given that CR = BP so we get

PQ = CR ……. (1)

In the △ PMQ and △ CMR we know that PQ || AC and QR is the transversal

We know that ∠ PQM and ∠ CRM are alternate angles and ∠ PMQ and ∠ CMR are vertically opposite angles

∠ PQM = ∠ CRM

∠ PMQ = ∠ CMR

By considering equation (1) and AAS congruence criterion

△ PMQ ≅ △ CMR

We know that the corresponding parts of congruent triangles are equal

PM = MC

Therefore, it is proved that QR bisects PC.