Difference between revisions of "2010 AIME II Problems/Problem 12"
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− | <math>\begin{array}{ | + | <math>\begin{array}{cccl} |
7c(\sqrt{a^2-(7c)^2})&=&8c(\sqrt{b^2-(8c)^2})&{}\\ | 7c(\sqrt{a^2-(7c)^2})&=&8c(\sqrt{b^2-(8c)^2})&{}\\ | ||
7(\sqrt{(a+7c)(a-7c)})&=&8(\sqrt{b+8c)(b-8c)})&{}\\ | 7(\sqrt{(a+7c)(a-7c)})&=&8(\sqrt{b+8c)(b-8c)})&{}\\ | ||
− | 7(\sqrt{(a-7c)})&=&8(\sqrt{(b-8c)})&\text{(Note that} a+7c=b+8c)\\ | + | 7(\sqrt{(a-7c)})&=&8(\sqrt{(b-8c)})&\text{(Note that } a+7c=b+8c)\\ |
49a-343c&=&64b-512c&{}\\ | 49a-343c&=&64b-512c&{}\\ | ||
49a+169c&=&64b&{}\\ | 49a+169c&=&64b&{}\\ | ||
− | 49a+169(a-b)&=&64b&\text{(Note that} c=a-b)\\ | + | 49a+169(a-b)&=&64b&\text{(Note that } c=a-b)\\ |
218a&=&233b&{}\\ | 218a&=&233b&{}\\ | ||
\end{array}</math> | \end{array}</math> |
Revision as of 20:20, 13 March 2015
Contents
Problem
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is . Find the minimum possible value of their common perimeter.
Solution 1
Let the first triangle have side lengths , , , and the second triangle have side lengths , , , where .
Equal perimeter:
Equal Area:
Since and are integer, the minimum occurs when , , and . Hence, the perimeter is .
Solution 2
Let be the semiperimeter of the two triangles. Also, let the base of the longer triangle be and the base of the shorter triangle be for some arbitrary factor . Then, the dimensions of the two triangles must be and . By Heron's Formula, we have
Since and are coprime, to minimize, we must have and . However, we want the minimum perimeter. This means that we must multiply our minimum semiperimeter by , which gives us a final answer of .
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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.