2002 AMC 12B Problems/Problem 23: Difference between revisions
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\qquad\mathrm{(D)}\ \frac 32 | \qquad\mathrm{(D)}\ \frac 32 | ||
\qquad\mathrm{(E)}\ \sqrt{3}</math> | \qquad\mathrm{(E)}\ \sqrt{3}</math> | ||
== Solution == | == Solution == | ||
Revision as of 22:08, 26 November 2017
Problem
In 
, we have 
and 
. Side 
and the median from 
to have the same length. What is 
?
Solution
Solution 1
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
![\[5 = 10a^2\]](http://latex.artofproblemsolving.com/b/7/e/b7ef5bf2a35df8f95e354145fc0ffc141e39ae7a.png)
Hence 
and 
.
Solution 2
From Stewart's Theorem, we have 
Simplifying, we get 
Solution 3
Let be the foot of the altitude from to extended past 
. Let 
be 
and 
be 
.
Using the Pythagorean Theorem, we obtain the equations
Subtracting the first from the second and third equations, we get
Then subtracting two times the first equation from the second and rearranging, we get 
, so 
Solution 2
From Stewart's Theorem, we have Simplifying, we get
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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