Question

Ramesh and Reena are playing with triangle ABC. Ramesh draws a line that bisects $\angle BAC$ ; this line cuts BC at D. Reena then extends AD to a point P. In response, Ramesh joins B and P. Reena then announces that BD bisects $\angle PBA$ , what a surprise! Together, Ramesh and Reena find that $BD=6\text{ cm}$ , $AC=9\text{ cm}$ , $DC=5\text{ cm}$ , $BP=8\text{ cm}$ , and $DP=5\text{ cm}$ .

How long is AP?

Accepted Answer

B. By Angle Bisector Theorem in $	riangle ABC$: $\frac{AB}{BD} = \frac{AC}{DC} \implies \frac{AB}{6} = \frac{9}{5} \implies AB = 10.8	ext{ cm}$. In $	riangle PBA$, BD bisects $\angle PBA$: $\frac{PB}{PD} = \frac{AB}{AD} \implies \frac{8}{5} = \frac{10.8}{AD} \implies AD = 6.75	ext{ cm}$. $AP = AD + DP = 6.75 + 5 = 11.75	ext{ cm}$.

Triangle Angle Bisectors

Explanation

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