Convex quadrilateral ABCD has AB = 3, BC = 4, CD = 13, AD = 12, and ∠ ABC = 90°, as shown. What is the area of the quadrilateral?

2014 AMC 12B Problem 9

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Convex quadrilateral

ABCD

has

AB = 3,

BC = 4,

CD = 13,

AD = 12,

and

\angle ABC = 90^\circ,

as shown. What is the area of the quadrilateral?

30

36

40

48

58.5

Use right triangle

ABC

to find diagonal

AC

Check whether

AC,

AD,

CD

satisfy

AC^2 + AD^2 = CD^2

By the Pythagorean Theorem in right triangle

ABC,

AC = \sqrt{3^2+4^2} = 5.

Since

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

the converse of the Pythagorean Theorem shows

\angle DAC = 90^\circ,

\triangle DAC

is right.

The area of

\triangle ABC

\tfrac12\cdot3\cdot4 = 6

and the area of

\triangle DAC

\tfrac12\cdot5\cdot12 = 30.

The quadrilateral has area

6 + 30 = 36.

Thus, the correct answer is B.