2015 AMC 12B Problems/Problem 19

Problem

In $\triangle ABC$ , $\angle C = 90^\circ$ and $AB = 12$ . Squares $ABXY$ and $ACWZ$ 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 1

First, we should find the center and radius of this circle. We can find the center by drawing the perpendicular bisectors of $WZ$ and $XY$ and finding their intersection point. This point happens to be the midpoint of $AB$ , the hypotenuse. Let this point be $M$ . To find the radius, determine $MY$ , where $MY^{2} = MA^2 + AY^2$ , $MA = \frac{12}{2} = 6$ , and $AY = AB = 12$ . Thus, the radius $=r =MY = 6\sqrt5$ .

Next we let $AC = b$ and $BC = a$ . Consider the right triangle $ACB$ first. Using the pythagorean theorem, we find that $a^2 + b^2 = 12^2 = 144$ . Next, we let $M'$ to be the midpoint of $WZ$ , and we consider right triangle $ZM'M$ . By the pythagorean theorem, we have that $\left(\frac{b}{2}\right)^2 + \left(b + \frac{a}{2}\right)^2 = r^2 = 180$ . Expanding this equation, we get that

$\frac{1}{4}(a^2+b^2) + b^2 + ab = 180$ $\frac{144}{4} + b^2 + ab = 180$ $b^2 + ab = 144 = a^2 + b^2$ $ab = a^2$ $b = a$

This means that $ABC$ is a 45-45-90 triangle, so $a = b = \frac{12}{\sqrt2} = 6\sqrt2$ . Thus the perimeter is $a + b + AB = 12\sqrt2 + 12$ which is answer $\boxed{\textbf{(C)}\; 12 + 12\sqrt2}$ .

Solution 2

The center of the circle on which $X$ , $Y$ , $Z$ , and $W$ lie must be equidistant from each of these four points. Draw the perpendicular bisectors of $\overline{XY}$ and of $\overline{WZ}$ . Note that the perpendicular bisector of $\overline{XY}$ is parallel to $\overline{BX}$ and passes through the midpoint of $\overline{AC}$ . Therefore, the triangle that is formed by $A$ , the midpoint of $\overline{AC}$ , and the point at which this perpendicular bisector intersects $\overline{AB}$ must be similar to $\triangle ABC$ , and the ratio of a side of the smaller triangle to a side of $\triangle ABC$ is 1:2. Consequently, the perpendicular bisector of $\overline{XY}$ passes through the midpoint of $\overline{AB}$ . The perpendicular bisector of $\overline{WZ}$ must include the midpoint of $\overline{AB}$ as well. Since all points on a perpendicular bisector of any two points $M$ and $N$ are equidistant from $M$ and $N$ , the center of the circle must be the midpoint of $\overline{AB}$ .

Now the distance between the midpoint of $\overline{AB}$ and $Z$ , which is equal to the radius of this circle, is $\sqrt{12^2 + 6^2} = \sqrt{180}$ . Let $a=AC$ . Then the distance between the midpoint of $\overline{AB}$ and $Y$ , also equal to the radius of the circle, is given by $\sqrt{\left(\frac{a}{2}\right)^2 + \left(a + \frac{\sqrt{144 - a^2}}{2}\right)^2}$ (the ratio of the similar triangles is involved here). Squaring these two expressions for the radius and equating the results, we have

$\left(\frac{a}{2}\right)^2+\left(a+\frac{\sqrt{144-a^2}}{2}\right)^{2} = 180$ $144 - a^2 = a\sqrt{144-a^2}$ $(144-a^2)^2 = a^2(144-a^2)$

Since $a$ cannot be equal to 12, the length of the hypotenuse of the right triangle, we can divide by $(144-a^2)$ , and arrive at $a = 6\sqrt{2}$ . The length of other leg of the triangle must be $\sqrt{144-72} = 6\sqrt{2}$ . Thus, the perimeter of the triangle is $12+2(6\sqrt{2}) = \boxed{\textbf{(C)}\; 12+12\sqrt{2}}$ .

2015 AMC 12B (Problems • Answer Key • Resources)
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2015 AMC 12B Problems/Problem 19

Contents

Problem

Solution 1

Solution 2

See Also