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

Question

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

trazonombresrg · Accepted Answer

Step 1The 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 2Then                     P        Q                    A        B              =                    P        F                    A        F              =            2              13              .  P  Q  =            2              13              ⋅  A  B  =            30              13              .

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.

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