1999 AIME Problem 2

Question

Consider the parallelogram with vertices

(10, 45),

(10, 114),

(28, 153),

and

(28, 84).

A line through the origin cuts this figure into two congruent polygons. The slope of the line is

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

Answer: 118

Concepts:parallelogram midpoint slope symmetry

Difficulty rating: 1790

Small hint:

A line cuts a parallelogram into two congruent pieces when it passes through the parallelogram’s center of symmetry

Big hint:

The center is the midpoint of the diagonal from

(10, 45)

to

(28, 153)

Solution:

Congruent pieces have equal area. For any fixed direction, there is only one line in that direction that bisects the area of a convex figure; because a parallelogram is centrally symmetric, that line passes through its center. Thus the required line through the origin must also pass through the center. Conversely, the

180^\circ

rotation about the center swaps the two pieces cut by such a line, so they are congruent. The center is the midpoint of a diagonal:

\begin{aligned} &\left(\frac{10 + 28}{2}, \frac{45 + 153}{2}\right) \\ &= (19, 99). \end{aligned}

The line through the origin and

(19, 99)

has slope

\frac{99}{19},

and

\gcd(99, 19) = 1,

so

m + n = 99 + 19 = 118.