Difference between revisions of "2010 AMC 12A Problems/Problem 14"
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== Solution == | == Solution == | ||
− | By the [[Angle Bisector Theorem]], we know that <math>\frac{AB}{ | + | By the [[Angle Bisector Theorem]], we know that <math>\frac{AB}{BC} = \frac{3}{8}</math>. If we use the lowest possible integer values for AB and BC (the measures of AD and DC, respectively), then <math>AB + BC = AD + DC = AC</math>, contradicting the [[Triangle Inequality]]. If we use the next lowest values (<math>AB = 6</math> and <math>BC = 16</math>), the Triangle Inequality is satisfied. Therefore, our answer is <math>6 + 16 + 3 + 8 = \boxed{33}</math>, or choice <math>\textbf{(B)}</math>. |
==Video Solution== | ==Video Solution== |
Latest revision as of 00:40, 20 October 2020
- The following problem is from both the 2010 AMC 12A #14 and 2010 AMC 10A #16, so both problems redirect to this page.
Contents
Problem
Nondegenerate has integer side lengths, is an angle bisector, , and . What is the smallest possible value of the perimeter?
Solution
By the Angle Bisector Theorem, we know that . If we use the lowest possible integer values for AB and BC (the measures of AD and DC, respectively), then , contradicting the Triangle Inequality. If we use the next lowest values ( and ), the Triangle Inequality is satisfied. Therefore, our answer is , or choice .
Video Solution
~IceMatrix
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.