Japan 2007

Question

A quadrilateral ABCD that satisfies AB = 5, BC = 7, CD = 6 is given. And AC and BD are perpendicular to each other. Find the length of DA.

Accepted Answer

Define P as the intersection point of AC and BD. Then from the Pythagorean theorem, AB^2 = AP^2 + BP^2, BC^2 = BP^2 + CP^2, CD^2 = CP^2 + DP^2, DA^2 = DP^2 + AP^2. Then DA^2 = AB^2 + CD^2 - BC^2 = 5^2 + 6^2 - 7^2 = 12. So DA = 23.