Question

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?

$30$

$36$

$40$

$48$

$58.5$

Answer: B

Concepts:Pythagorean Theorem triangle area area decomposition

Difficulty rating: 1560

Solution:

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 $\angle DAC = 90^\circ,$ so $\triangle DAC$ is right.

The area of $\triangle ABC$ is $\tfrac12\cdot3\cdot4 = 6$ and the area of $\triangle DAC$ is $\tfrac12\cdot5\cdot12 = 30.$ The quadrilateral has area $6 + 30 = 36.$

Thus, the correct answer is B.

2014 AMC 12B Problem 9

Answer: B

Difficulty rating: 1560

Solution:

Problem 9 in Other Years