1995 AMC 12 Problem 22

Question

A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths

13,

19,

20,

25

and

31,

although this is not necessarily their order around the pentagon. The area of the pentagon is

459

600

680

720

745

Answer: E

Concepts:rectangles Pythagorean theorem area subtraction

Difficulty rating: 1910

Small hint:

Two pentagon sides are the original rectangle’s full side lengths

Big hint:

Look for two differences among the lengths that form the legs of a right triangle with a third listed length

Solution:

The full rectangle sides must be longer than the two remnants on those same sides. The only assignment whose two differences and remaining cut side form a right triangle is the rectangle

25

by

31,

with remnants

20

and

19.

The removed corner then has legs

25-20=5

and

31-19=12,

whose hypotenuse is the listed side

13.

Thus the pentagon’s area is

\begin{aligned} 25\cdot31-\frac12(5)(12) &=775-30\\ &=745. \end{aligned}

Hence the correct answer is E.