Difference between revisions of "2020 AMC 8 Problems/Problem 18"
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− | + | Rectangle <math>ABCD</math> is inscribed in a semicircle with diameter <math>\overline{FE},</math> as shown in the figure. Let <math>DA=16,</math> and let <math>FD=AE=9.</math> What is the area of <math>ABCD?</math> | |
+ | |||
+ | <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> | ||
+ | <math>\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272</math> | ||
+ | |||
+ | ==Solution== | ||
+ | First, realize <math>ABCD</math> is not a square. It can easily be seen that the diameter of the semicircle is <math>9+16+9=34</math>, so the radius is <math>\frac{34}{2}=17</math>. Express the area of Rectangle <math>ABCD</math> as <math>16h</math>, where <math>h=AB</math>. Notice that by the Pythagorean theorem <math>8^2+h^{2}=17^{2}\implies h=15</math>. Then, the area of Rectangle <math>ABCD</math> is equal to <math>\boxed{\textbf{(A) }240}</math>. ~icematrix | ||
+ | |||
+ | {{AMC8 box|year=2020|num-b=17|num-a=19}} | ||
+ | |||
+ | [[Category:Introductory Geometry Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 00:56, 18 November 2020
Rectangle is inscribed in a semicircle with diameter
as shown in the figure. Let
and let
What is the area of
Solution
First, realize is not a square. It can easily be seen that the diameter of the semicircle is
, so the radius is
. Express the area of Rectangle
as
, where
. Notice that by the Pythagorean theorem
. Then, the area of Rectangle
is equal to
. ~icematrix
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AJHSME/AMC 8 Problems and Solutions |
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