Difference between revisions of "2022 AMC 10A Problems/Problem 15"
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
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When we look at the sidelengths of quadrilateral we see 7 and 24, which screams out 25 because of Pythagorean triplets. As a result, we can draw a line through points <math>A</math> and <math>C</math> to make a diameter of <math>25</math>. Since the diameter is 25, we can see the area of the circle is just <math>\frac{625\pi}{4}</math> from the formula doe the area of the circle with just a diameter. Then using Brahmagupta Formula, | When we look at the sidelengths of quadrilateral we see 7 and 24, which screams out 25 because of Pythagorean triplets. As a result, we can draw a line through points <math>A</math> and <math>C</math> to make a diameter of <math>25</math>. Since the diameter is 25, we can see the area of the circle is just <math>\frac{625\pi}{4}</math> from the formula doe the area of the circle with just a diameter. Then using Brahmagupta Formula, |
Revision as of 14:47, 13 November 2022
Problem
Quadrilateral with side lengths is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form where and are positive integers such that and have no common prime factor. What is
Solution 1
DIAGRAM IN PROGRESS.
WILL BE DONE TOMORROW, WAIT FOR ME THANKS.
Opposite angles of every cyclic quadrilateral are supplementary, so We claim that We can prove it by contradiction:
- If then and are both acute angles. This arrives at a contradiction.
- If then and are both obtuse angles. This arrives at a contradiction.
By the Inscribed Angle Theorem, we conclude that is the diameter of the circle. So, the radius of the circle is
The area of the requested region is Therefore, the answer is
~MRENTHUSIASM
Solution 2 (Brahmagupta Formula)
When we look at the sidelengths of quadrilateral we see 7 and 24, which screams out 25 because of Pythagorean triplets. As a result, we can draw a line through points and to make a diameter of . Since the diameter is 25, we can see the area of the circle is just from the formula doe the area of the circle with just a diameter. Then using Brahmagupta Formula,
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 10 Problems and Solutions |
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