Let triangle ABC have AB=15, BC=14, and CA=13. Let ℓ_1 be the angle bisector of angle ABC and let ℓ_2 be the angle bisector of angle BAC. If the projection of B onto ℓ_2 is Q and the projection of A onto ℓ_1 is P find PQ.

granuliet1u

granuliet1u

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2022-08-17

Let A B C have A B = 15, B C = 14, and C A = 13.. Let 1 be the angle bisector of A B C and let 2 be the angle bisector of B A C .. If the projection of B onto 2 is Q and the projection of A onto 1 is P find PQ.

Answer & Explanation

trazonombresrg

trazonombresrg

Beginner2022-08-18Added 8 answers

Step 1
The similarity ratio is P F A F = cos A F P ..
Then P F A F = cos A F P = cos ( 1 2 ( B A C + C B A ) ) = cos ( 1 2 ( π A C B ) ) = sin 1 2 A C B = 1 cos A C B 2 = 1 ( A C 2 + B C 2 A B 2 2 A C B C ) 2 = 2 13 .
Step 2
Then P Q A B = P F A F = 2 13 .
P Q = 2 13 A B = 30 13 .

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