MathProblem 43 Wall and two houses problem

There is a castle surrounded by a circular wall. The wall has a north gate and a south gate. A yellow house lays 3 miles north of the north gate. A blue house lays 9 miles east of the south gate. What is the radius of the circle formed by the wall?

Solution

这里没有提到一个条件是，两个房子的连线与圆相切。所以不妨假设半径为 \(R\). 设黄色房子那里的夹角为 \(\alpha\), 那么根据正切值：

\[\tan(\alpha) = \frac{9}{3+2R}=\frac{R}{\sqrt{6R+9}} \]

求解 \(R\) 即可