Problem 145

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Problem 145: (2015 AMC 10b, 19; 12b 19)

In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle? $\textbf{(A)}\; 12+9\sqrt{3} \qquad$ $\textbf{(B)}\; 18+6\sqrt{3} \qquad$ $\textbf{(C)}\; 12+12\sqrt{2} \qquad$ $\textbf{(D)}\; 30 \qquad$ $\textbf{(E)}\; 32$

Solution

Exercise 145: Solution

Equilateral triangles $ABX$ and $BCY$ are constructed outside $\triangle ABC$, on sides $AB$ and $BC$ respectively. Points $X$, $Y$, $C$, $A$ lie on a circle. If $AB=3$ and $BC=5$, what is $CA$?

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