Difference between revisions of "2020 AMC 8 Problems/Problem 18"

(Solution)
(Solution 2)
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==Solution 2==
 
==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("<math>A</math>",(8,0), 1.25*S); dot("<math>B</math>",(8,15), 1.25*N); dot("<math>C</math>",(-8,15), 1.25*N); dot("<math>D</math>",(-8,0), 1.25*S); dot("<math>E</math>",(17,0), 1.25*S); dot("<math>F</math>",(-17,0), 1.25*S); label("<math>16</math>",(0,0),N); label("<math>9</math>",(12.5,0),N); label("<math>9</math>",(-12.5,0),N); dot("<math>O</math>", (0,0), 1.25*N);[/asy]
+
<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*N);</asy>
  
 
We have <math>OC=17</math>, as it is a radius, and <math>OD=8</math> since it is half of <math>AD</math>. This means that <math>CD=\sqrt{17^2-8^2}=15</math>. So <math>16*15=\boxed{\textbf{(A)}240}</math>
 
We have <math>OC=17</math>, as it is a radius, and <math>OD=8</math> since it is half of <math>AD</math>. This means that <math>CD=\sqrt{17^2-8^2}=15</math>. So <math>16*15=\boxed{\textbf{(A)}240}</math>

Revision as of 02:37, 18 November 2020

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*N);[/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

See also

2020 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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