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Triangle ABC is a right triangle with ∠BAC = 90° and ∠B > ∠C. If AP is an altitude of the triangle, AQ is an angle bisector of the triangle, and AR is a median of the triangle, and ∠PAQ = 13°. If P is on BQ, then what is the measure of ∠RAC?

 May 5, 2023
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Since ∠PAQ = 13°, then ∠QAP = 90 - 13 = 77°.

Since AQ is an angle bisector, then ∠BAQ = ∠QAP/2 = 38.5°.

Since AR is a median, then ∠BAR = ∠BAC/2 = 45°.

Since ∠B > ∠C, then ∠BAC = 90 - ∠C.

Therefore, ∠C = 45°.

Therefore, ∠RAC = ∠BAR - ∠BAQ = 45 - 38.5 = 6.5°.

 May 5, 2023

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