Difference between revisions of "1987 AIME Problems/Problem 9"
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=== Note === | === Note === | ||
This is the [[Fermat point]] of the triangle. | This is the [[Fermat point]] of the triangle. | ||
+ | |||
+ | == Video Solution by Pi Academy == | ||
+ | |||
+ | https://youtu.be/fZAChuJDlSw?si=wJUPmgVRlYwazauh | ||
+ | |||
+ | ~ smartschoolboy9 | ||
== See also == | == See also == |
Latest revision as of 12:37, 11 December 2024
Problem
Triangle has right angle at , and contains a point for which , , and . Find .
Solution
Let . Since , each of them is equal to . By the Law of Cosines applied to triangles , and at their respective angles , remembering that , we have
Then by the Pythagorean Theorem, , so
and
Note
This is the Fermat point of the triangle.
Video Solution by Pi Academy
https://youtu.be/fZAChuJDlSw?si=wJUPmgVRlYwazauh
~ smartschoolboy9
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.