2010 AMC 10B Problems/Problem 19
Problem
A circle with center has area . Triangle is equilateral, is a chord on the circle, , and point is outside . What is the side length of ?
Solution 1
The formula for the area of a circle is so the radius of this circle is
Because must be in the interior of circle
Let be the side length of the triangle, the unknown value, and let be the point on where Since is equilateral, and We are given Use the Pythagorean Theorem and solve for
Solution 2
We can use the same diagram as Solution 1 and label the side length of as . Using congruent triangles, namely the two triangles and , we get that . From this, we can use the Law of Cosines, to get Simplifying, we get We can factor this to get Lengths must be non-negative, so the answer is ~bryan gao
Video Solution
https://youtu.be/FQO-0E2zUVI?t=906
~IceMatrix
See Also
2010 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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All AMC 10 Problems and Solutions |
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