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Question:

Ramesh and Reena are playing with triangle ∠ABC. Ramesh draws a line that bisects ; 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 ∠PBA, hat a surprise! Together, Ramesh and Reena find that BD = 6 cm, AC = 9 cm, DC = 5 cm, BP = 8 cm, and DP = 5 cm.

How long is AP?

Explanation:

Given:
BD = 6 cm, AC = 9 cm, DC = 5 cm, BP = 8 cm, and DP = 5 cm.
Since AD is the angular bisector applying the angular bisector theorem we have:

$\frac{A B}{B D}$ = $\frac{A C}{C D}$

Hence : Considering AB = x cm.

$\frac{9}{5}$ = $\frac{x}{6}$

x = 10.8 cm.
Now since BD is the angular bisector for angle PBA we have :
Applyinh the internal angle bisector theorem:

$\frac{P B}{P D}$ = $\frac{B A}{A D}$

Considering AD = y cm.

$\frac{8}{5}$ = $\frac{10.8}{y}$

y = 6.75 cm.
AP = AD + DP.
= 6.75 + 5 = 11.75 cm

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