Difference between revisions of "2018 AMC 12B Problems/Problem 4"
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==Problem== | ==Problem== | ||
− | A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle? | + | A circle has a chord of length <math>10</math>, and the distance from the center of the circle to the chord is <math>5</math>. What is the area of the circle? |
− | (A) | + | <math> |
+ | \textbf{(A) }25\pi \qquad | ||
+ | \textbf{(B) }50\pi \qquad | ||
+ | \textbf{(C) }75\pi \qquad | ||
+ | \textbf{(D) }100\pi \qquad | ||
+ | \textbf{(E) }125\pi \qquad | ||
+ | </math> | ||
==Solution== | ==Solution== | ||
+ | Let <math>O</math> be the center of the circle, <math>\overline{AB}</math> be the chord, and <math>M</math> be the midpoint of <math>\overline{AB},</math> as shown below. | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(200); | ||
+ | pair O, A, B, M; | ||
+ | O = (0,0); | ||
+ | A = (-5,5); | ||
+ | B = (5,5); | ||
+ | M = midpoint(A--B); | ||
+ | draw(Circle(O,5sqrt(2))); | ||
+ | dot("$O$", O, 1.5*S, linewidth(4.5)); | ||
+ | dot("$A$", A, 1.5*NW, linewidth(4.5)); | ||
+ | dot("$B$", B, 1.5*NE, linewidth(4.5)); | ||
+ | dot("$M$", M, 1.5*N, linewidth(4.5)); | ||
+ | draw(A--B^^M--O^^A--O^^M--O^^B--O); | ||
+ | label("$5$", midpoint(A--M), 1.5*N); | ||
+ | label("$5$", midpoint(B--M), 1.5*N); | ||
+ | label("$5$", midpoint(O--M), 1.5*E); | ||
+ | label("$r$", midpoint(O--A), 1.5*SW); | ||
+ | label("$r$", midpoint(O--B), 1.5*SE); | ||
+ | </asy> | ||
+ | Note that <math>\overline{OM}\perp\overline{AB}.</math> Since <math>OM=AM=BM=5,</math> we conclude that <math>\triangle OMA</math> and <math>\triangle OMB</math> are congruent isosceles right triangles. It follows that <math>r=5\sqrt2,</math> so the area of <math>\odot O</math> is <math>\pi r^2=\boxed{\textbf{(B) }50\pi}</math>. | ||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | ==Video Solution (HOW TO THINK CRITICALLY!!!)== | ||
+ | https://youtu.be/bJvXMkwjrtE | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==See Also== | ||
{{AMC12 box|year=2018|ab=B|num-b=3|num-a=5}} | {{AMC12 box|year=2018|ab=B|num-b=3|num-a=5}} | ||
+ | {{MAA Notice}} | ||
+ | |||
+ | [[Category:Introductory Geometry Problems]] |
Latest revision as of 22:06, 27 May 2023
Problem
A circle has a chord of length , and the distance from the center of the circle to the chord is
. What is the area of the circle?
Solution
Let be the center of the circle,
be the chord, and
be the midpoint of
as shown below.
Note that
Since
we conclude that
and
are congruent isosceles right triangles. It follows that
so the area of
is
.
~MRENTHUSIASM
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.