Question

A triangle has vertices $P=(-8,5),$ $Q=(-15,-19),$ and $R=(1,-7).$ The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0.$ Find $a+c.$

Answer

Answer: 89

Concepts:angle bisector coordinate geometry vector

Difficulty rating: 2100

Small Hint:

Find the unit vectors from $P$ toward $Q$ and $R$

Big Hint:

The internal angle-bisector direction is the sum of the two unit vectors

Solution:

We have $\overrightarrow{PQ}=(-7,-24)$ with length $25,$ and $\overrightarrow{PR}=(9,-12)$ with length $15.$ The sum of their unit vectors is $\begin{aligned}v&=\left(-\frac7{25},-\frac{24}{25}\right)\\&\quad+\left(\frac35,-\frac45\right)\\&=\frac1{25}(8,-44),\end{aligned}$ so the angle bisector has direction $(2,-11).$ A normal vector is $(11,2).$ Through $P=(-8,5),$ its equation is $11(x+8)+2(y-5)=0,$ or $11x+2y+78=0.$ Hence $a+c=11+78=89.$

1990 AIME Problem 7

Answer: 89

Difficulty rating: 2100

Small Hint:

Big Hint:

Solution:

Problem 7 in Other Years