Difference between revisions of "2008 AMC 10A Problems/Problem 18"
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=== Solution 2 === | === Solution 2 === | ||
From the formula <math>A = rs</math>, where <math>A</math> is the area of a triangle, <math>r</math> is its [[inradius]], and <math>s</math> is the [[semiperimeter]], we can find that <math>r = \frac{20}{32/2} = \frac{5}{4}</math>. It is known that in a right triangle, <math>r = s - h</math>, where <math>h</math> is the hypotenuse, so <math>h = 16 - \frac{5}{4} = \frac{59}{4}</math>. | From the formula <math>A = rs</math>, where <math>A</math> is the area of a triangle, <math>r</math> is its [[inradius]], and <math>s</math> is the [[semiperimeter]], we can find that <math>r = \frac{20}{32/2} = \frac{5}{4}</math>. It is known that in a right triangle, <math>r = s - h</math>, where <math>h</math> is the hypotenuse, so <math>h = 16 - \frac{5}{4} = \frac{59}{4}</math>. | ||
+ | |||
+ | === Solution 3 === | ||
+ | From the problem, we know that | ||
+ | <center><math>\begin{align*} | ||
+ | a+b+c &= 32 \\ | ||
+ | 2ab &= 80. \\ | ||
+ | \end{align*}</math></center> | ||
+ | |||
+ | Subtracting <math>c</math> from both sides of the first equation and squaring both sides, we get | ||
+ | <center><math>\begin{align*} | ||
+ | (a+b)^2 &= (32 - c)^2\\ | ||
+ | a^2 + b^2 + 2ab &= 32^2 + c^2 - 64c.\\ | ||
+ | \end{align*}</math></center> | ||
+ | |||
+ | Now we substitute in <math>a^2 + b^2 = c^2</math> as well as <math>2ab = 80</math> into the equation to get | ||
+ | <center><math>\begin{align*} | ||
+ | 80 &= 1024 - 64c\\ | ||
+ | c &= \frac{944}{64}. | ||
+ | \end{align*}</math></center> | ||
+ | |||
+ | Further simplification yields the result of <math>\frac{59}{4}</math>. | ||
==See also== | ==See also== |
Revision as of 18:34, 16 December 2011
Problem
A right triangle has perimeter and area . What is the length of its hypotenuse?
Solution
Solution 1
Let the legs of the triangle have lengths . Then, by the Pythagorean Theorem, the length of the hypotenuse is , and the area of the triangle is . So we have the two equations
a+b+\sqrt{a^2+b^2} &= 32 \\ \frac{1}{2}ab &= 20
\end{align}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align} allowed only in paragraph mode.)Re-arranging the first equation and squaring,
\sqrt{a^2+b^2} &= 32-(a+b)\\ a^2 + b^2 &= 32^2 - 64(a+b) + (a+b)^2\\ a^2 + b^2 + 64(a+b) &= a^2 + b^2 + 2ab + 32^2\\
a+b &= \frac{2ab+32^2}{64}\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)From we have , so
The length of the hypotenuse is .
Solution 2
From the formula , where is the area of a triangle, is its inradius, and is the semiperimeter, we can find that . It is known that in a right triangle, , where is the hypotenuse, so .
Solution 3
From the problem, we know that
a+b+c &= 32 \\ 2ab &= 80. \\
\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)Subtracting from both sides of the first equation and squaring both sides, we get
(a+b)^2 &= (32 - c)^2\\ a^2 + b^2 + 2ab &= 32^2 + c^2 - 64c.\\
\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)Now we substitute in as well as into the equation to get
80 &= 1024 - 64c\\ c &= \frac{944}{64}.
\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)Further simplification yields the result of .
See also
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 10 Problems and Solutions |