Difference between revisions of "2002 AMC 12B Problems/Problem 23"
(→Solution) |
(redirect link to median of a triangle instead of median of a set.) |
||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | In <math>\triangle ABC</math>, we have <math>AB = 1</math> and <math>AC = 2</math>. Side <math>\overline{BC}</math> and the [[median]] from <math>A</math> to <math>\overline{BC}</math> have the same length. What is <math>BC</math>? | + | In <math>\triangle ABC</math>, we have <math>AB = 1</math> and <math>AC = 2</math>. Side <math>\overline{BC}</math> and the [[Median of a triangle|median]] from <math>A</math> to <math>\overline{BC}</math> have the same length. What is <math>BC</math>? |
<math>\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} | <math>\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} |
Revision as of 17:01, 25 February 2008
Problem
In , we have and . Side and the median from to have the same length. What is ?
Solution
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 .
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 |