2021 JMPSC Sprint Problems/Problem 16

Problem

$ABCD$ is a concave quadrilateral with $AB = 12$ , $BC = 16$ , $AD = CD = 26$ , and $\angle ABC=90^\circ$ . Find the area of $ABCD$ .

Solution

Notice that $[ABCD] = [ADC] - [ABC]$ and $AC = \sqrt{12^2 + 16^2} = 20$ by the Pythagorean Thereom. We then have that the area of triangle of $ADC$ is $\frac{20 \cdot \sqrt{26^2 - 10^2}}{2} = 240$ , and the area of triangle $ABC$ is $\frac{12 \cdot 16}{2} = 96$ , so the area of quadrilateral $ABCD$ is $240 - 96 = 144$ .