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In ∆PQR, ∠QIR – ∠POQ = 26° where I and O are the incentre and orthocentre of that triangle. If the measure of ∠Q of the triangle is 28° and C is the circle-center of the triangle, then what is the difference between the measure of ∠PIQ and ∠QCR?

In ∆PQR, ∠QIR – ∠POQ = 26° where I and O are the incentre and orthocentre of that triangle. If the measure of ∠Q of the triangle is 28° and C is the circle-center of the triangle, then what is the difference between the measure of ∠PIQ and ∠QCR? Correct Answer 14°

Given:

∠QIR – ∠POQ = 26°

∠Q = 28°

Calculation:

∠P + ∠Q + ∠R = 180°

⇒ ∠P + 28° + ∠R = 180°

⇒ ∠P + ∠R = 152° ----(i)

∠QIR = 90° + (∠P/2)

∠POQ = 180° – ∠R

∵ ∠QIR – ∠POQ = 26°

⇒ 90° + (∠P/2) – 180° + ∠R = 26°

⇒ ∠P + 2∠R = 232°      ----(ii)
From (i) and (ii), we get

∠P = 72° and ∠R = 80°

∠PIQ = 90° + (∠R/2)

⇒ ∠PIQ = 90° + 40° = 130°

∠QCR = 2 × ∠P

⇒ ∠QCR = 2 × 72° = 144°

∴ Required difference = 144° – 130°

= 14°

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