2020 AMC 8 Problems/Problem 18

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$

$[asy] draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("A",(8,0), 1.25*S); dot("B",(8,15), 1.25*N); dot("C",(-8,15), 1.25*N); dot("D",(-8,0), 1.25*S); dot("E",(17,0), 1.25*S); dot("F",(-17,0), 1.25*S); label("16",(0,0),N); label("9",(12.5,0),N); label("9",(-12.5,0),N); [/asy]$ $\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272$

Solution

First, realize $ABCD$ is not a square. It can easily be seen that the diameter of the semicircle is $9+16+9=34$, so the radius is $\frac{34}{2}=17$. Express the area of Rectangle $ABCD$ as $16h$, where $h=AB$. Notice that by the Pythagorean theorem $8^2+h^{2}=17^{2}\implies h=15$. Then, the area of Rectangle $ABCD$ is equal to $16\cdot 15=\boxed{\textbf{(A) }240}$. ~icematrix

Solution 2

$[asy] draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("A",(8,0), 1.25*S); dot("B",(8,15), 1.25*N); dot("C",(-8,15), 1.25*N); dot("D",(-8,0), 1.25*S); dot("E",(17,0), 1.25*S); dot("F",(-17,0), 1.25*S); label("16",(0,0),N); label("9",(12.5,0),N); label("9",(-12.5,0),N); dot("O", (0,0), 1.25*S); draw((0,0)--(-8,15));[/asy]$

We have $OC=17$, as it is a radius, and $OD=8$ since it is half of $AD$. This means that $CD=\sqrt{17^2-8^2}=15$. So $16*15=\boxed{\textbf{(A)}240}$

~yofro

Solution 3 (coordinate bashing)

Let the midpoint of segment $FE$ be the origin. Evidently, point $D$ is at $(-8, 0)$ and $A$ is at $(8, 0)$. Since points $C$ and $B$ share x-coordinates with $D$ and $A$, respectively, we can just find the y-coordinate of $B$ (which is just the width of the rectangle) and multiply this by $DA$, or $16$. Since the radius of the semicircle is $\frac{9+16+9}{2}$, or $17$, the equation of the circle that our semicircle is a part of is $x^2+y^2=289$. Since we know that the x-coordinate of $B$ is $8$, we can plug this into our equation to obtain that $y=\pm15$. Since $y>0$, as the diagram suggests, we know that the y-coordinate of $B$ is $15$. Therefore, our answer is $16\cdot 15$, or $\boxed{\textbf{(A) }240}$.

NOTE: The synthetic solution is definitely faster and more elegant. However, this is the solution that you should use if you can't see any other easier solution.

- StarryNight7210