Sides AB, BC, CD, and DA of convex polygon ABCD have lengths 3, 4, 12, and 13, respectively; and ∠ CBA is a right angle. The area of the quadrilateral is

1980 AMC 12 Problem 7

Attempt Problem 7 of the 1980 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1980 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Sides

AB,

BC,

CD,

and

DA

of convex polygon

ABCD

have lengths

3,

4,

12,

and

13,

respectively; and

\angle CBA

is a right angle. The area of the quadrilateral is

32

36

39

42

48

Draw diagonal

AC

and find its length from

\triangle ABC

Recognize a second right triangle using the side lengths

5,

12,

and

13

Triangle

ABC

is a

3

4

5

right triangle, so

AC=5

and its area is

6.

Since

5^2+12^2=13^2,

triangle

ACD

is right at

C

and has area

(\frac{1}{2})(5)(12)=30.

The quadrilateral’s area is

6+30=36.

Therefore, the correct answer is B.