1990 AIME Problems/Problem 7

From AoPSWiki

Problem

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 $a+c_{}^{}$ .

Solution

Use the distance formula to determine the lengths of each of the sides of the triangle. We find that it has lengths of side $15,\ 20,\ 25$ , indicating that it is a $3-4-5$ right triangle. At this point, we just need to find another point that lies on the bisector of $\angle P$ .

Solution 1

Use the angle bisector theorem to find that the angle bisector of $\angle P$ divides $QR$ into segments of length $\frac{25}{x} = \frac{15}{20 -x} \Longrightarrow x = \frac{25}{2},\ \frac{15}{2}$ . It follows that $\frac{QP'}{RP'} = \frac{5}{3}$ , and so $P' = \left(\frac{5x_R + 3x_Q}{8},\frac{5y_R + 3y_Q}{8}\right) = (-5,-23/2)$ .

The desired answer is the equation of the line $PP'$ . $PP'$ has slope $\frac{-11}{2}$ , from which we find the equation to be $11x + 2y + 78 = 0$ . Therefore, $a+c = \boxed{089}$ .

Solution 2

Extend $PR$ to a point $S$ such that $PS = 25$ . This forms an isosceles triangle $PQS$ . The coordinates of $S$ , using the slope of $PR$ (which is $-4/3$ ), can be determined to be $(7,-15)$ . Since the angle bisector of $\angle P$ must touch the midpoint of $QS \Rightarrow (-4,-17)$ , we have found our two points. We reach the same answer of $11x + 2y + 78 = 0$ .

Solution 3

By the angle bisector theorem as in solution 1, we find that $QP' = 25/2$ . If we draw the right triangle formed by $Q, P',$ and the point directly to the right of $Q$ and below $P'$ , we get another $3-4-5 \triangle$ (since the slope of $QR$ is $3/4$ ). Using this, we find that the horizontal projection of $QP'$ is $10$ and the vertical projection of $QP'$ is $15/2$ .

Thus, the angle bisector touches $QR$ at the point $\left(-15 + 10, -19 + \frac{15}{2}\right) = \left(-5,-\frac{23}{2}\right)$ , from where we continue with the first solution.

1990 AIME (Problems • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Personal tools

Navigation

Search

Toolbox