2018 AMC 12B Problems/Problem 4
Revision as of 08:42, 20 September 2021 by MRENTHUSIASM (talk | contribs) (→Solution: Made the solution clearer with diagrams.)
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.
![[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]](http://latex.artofproblemsolving.com/a/9/8/a98a012a46954b990fd2592abb76aa48f7e6cef5.png)
Recall that 
Since 
we conclude that 
and 
are congruent isosceles right triangles. It follows that 
so the area of 
is 
.
~MRENTHUSIASM
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. 
