Track / Stage 4 / 197 of 340 #457 of 1964

Problem 457

AMC 12 late, AIME early

Geometry Difficulty 4.8 Find the answer

In a cyclic quadrilateral $ABCD$ , the lengths of the four sides in sequence are $AB=2, BC=7, CD=6, DA=9$ . Then the area of the quadrilateral is $\qquad$
(2010, National Junior High School Mathematics League, Jiangxi Province Preliminary)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Because $7^{2}+6^{2}=2^{2}+9^{2}$ , therefore, $BD$ is the diameter of the circumcircle of quadrilateral $ABCD$ , i.e., $\triangle ABD$ and $\triangle CBD$ are right triangles. The calculated area of the required shape is 30.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.