Difference between revisions of "2018 AMC 12B Problems/Problem 4"
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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 10, and the distance from the center of the circle to the chord is 5. 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== | ||
Revision as of 22:17, 22 February 2018
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?
Solution
The shortest segment that connects the center of the circle to a chord is the perpendicular bisector of the chord. Applying the Pythagorean theorem, we find that ![\[r^2 = 5^2 + 5^2 = 50\]](http://latex.artofproblemsolving.com/d/e/f/def9253c749de80deb042412eb5965c8558b8c5c.png)
The area of a triangle is 
, so the answer is 
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. 
