Difference between revisions of "2007 AMC 12B Problems/Problem 23"
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And this gives us <math>6</math> solutions <math>\Rightarrow \mathrm{(A)}</math>. | And this gives us <math>6</math> solutions <math>\Rightarrow \mathrm{(A)}</math>. | ||
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+ | ==Solution #2== | ||
+ | We will proceed by using the fact that <math>[ABC] = r\cdot s</math>, where <math>r</math> is the radius of the incircle and <math>s</math> is the semiperimeter (<math>s = \frac{p}{2}</math>). | ||
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+ | We are given <math>[ABC] = 3p = 6s \Rightarrow rs = 6s \Rightarrow r = 6</math>. | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2007|ab=B|num-b=22|num-a=24}} | {{AMC12 box|year=2007|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 13:43, 29 November 2014
Contents
Problem 23
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to times their perimeters?
Solution
Let and
be the two legs of the triangle.
We have .
Then .
We can complete the square under the root, and we get, .
Let and
, we have
.
After rearranging, squaring both sides, and simplifying, we have .
Putting back and
, and after factoring using
, we've got
.
Factoring 72, we get 6 pairs of and
And this gives us solutions
.
Solution #2
We will proceed by using the fact that , where
is the radius of the incircle and
is the semiperimeter (
).
We are given .
See Also
2007 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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