2002 AMC 12B Problems/Problem 23
Contents
Problem
In , we have and . Side and the median from to have the same length. What is ?
Solution
Solution 1: Pythagorean Theorem
Let be the foot of the altitude from to extended past . Let and . Using the Pythagorean Theorem, we obtain the equations
Subtracting equation from and , we get
Then, subtracting from and rearranging, we get , so
~greenturtle 11/28/2017
Solution 2: Law of Cosines
Let be the foot of the median from to , and we let . Then by the Law of Cosines on , we have
Since , we can add these two equations and get
Hence and .
Solution 3: Stewart's Theorem
From Stewart's Theorem, we have Simplifying, we get - awu2014
Solution 4: Pappus's Median Theorem
There is a theorem in geometry known as Pappus's Median Theorem. It states that if you have , and you draw a median from point to side (label this as ), then: . Note that is the length of side , is the length of side , and is length of side . Let . Then . Now, we can plug into the formula given above: , , , and . After some simple algebra, we find . Then, .
-Flames
Note: Pappus's Median Theorem is just a special case of Stewart's Theorem, with . ~Puck_0 aka Apollonius' Theorem - Orion 2010
Video Solution by TheBeautyofMath
~IceMatrix
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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 |
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