Difference between revisions of "2007 AMC 12B Problems/Problem 23"
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We have <math>\frac{1}{2}ab = 3(a+b+c)</math>. | We have <math>\frac{1}{2}ab = 3(a+b+c)</math>. | ||
− | Then <math>ab=6 | + | Then <math>ab=6 \left(a+b+\sqrt {a^2 + b^2}\right)</math>. |
− | We can complete the square under the root, and we get, <math>ab=6 | + | We can complete the square under the root, and we get, <math>ab=6 \left(a+b+\sqrt {(a+b)^2 - 2ab}\right)</math>. |
− | Let <math>ab=p</math> and <math>a+b=s</math>, we have <math>p=6 | + | Let <math>ab=p</math> and <math>a+b=s</math>, we have <math>p=6 \left(s+ \sqrt {s^2 - 2p}\right)</math>. |
After rearranging, squaring both sides, and simplifying, we have <math>p=12s-72</math>. | After rearranging, squaring both sides, and simplifying, we have <math>p=12s-72</math>. | ||
− | Putting back <math>a</math> and <math>b</math>, and after factoring using <math>SFFT</math>, we've got <math>(a-12) | + | Putting back <math>a</math> and <math>b</math>, and after factoring using <math>SFFT</math>, we've got <math>(a-12)(b-12)=72</math>. |
Revision as of 13:28, 1 January 2016
Contents
[hide]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 .
The incircle of breaks the triangle's sides into segments such that , and . Since ABC is a right triangle, one of , and is equal to its radius, 6. Let's assume .
The side lengths then become , and . Plugging into Pythagorean's theorem:
We can factor to arrive with pairs of solutions: and .
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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