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In a ∆PQR, ∠Q : ∠R = 6 : 5 and I and O are the incentre and orthocentre of ∆PQR whereas I1 is the incentre of ∆QIR. If ∠QI1R = 147°, then what is the ratio of the measure of ∠POR to the measure of ∠PIQ?

In a ∆PQR, ∠Q : ∠R = 6 : 5 and I and O are the incentre and orthocentre of ∆PQR whereas I1 is the incentre of ∆QIR. If ∠QI1R = 147°, then what is the ratio of the measure of ∠POR to the measure of ∠PIQ? Correct Answer 9 : 10

Given:

∠Q : ∠R = 6 : 5 and ∠QI1R = 147°

Calculation:

∠QI1R = 90° + (∠QIR/2)

⇒ 147° = 90° + (∠QIR/2)

⇒ ∠QIR = 114°

∠QIR = 90° + (∠P/2)

⇒ 114 ° = 90° + (∠P/2)

⇒ ∠P = 48°

Suppose, ∠Q = 6x and ∠R = 5x.

⇒ 48° + 6x + 5x = 180°

⇒ 11x = 132°

⇒ x = 12

∠Q = 6x = 72°

∠R = 5x = 60°

∠POR = 180° – ∠Q

⇒ ∠POR = 180° – 72° = 108°

∠PIQ = 90° + (∠R/2)

⇒ ∠PIQ = 90° + 30° = 120°

∴ Required ratio = 108 : 120

= 9 : 10

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