Difference between revisions of "2008 AMC 10A Problems/Problem 18"
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− | ==Problem== | + | == Problem == |
A [[right triangle]] has [[perimeter]] <math>32</math> and area <math>20</math>. What is the length of its [[hypotenuse]]? | A [[right triangle]] has [[perimeter]] <math>32</math> and area <math>20</math>. What is the length of its [[hypotenuse]]? | ||
<math>\mathrm{(A)}\ \frac{57}{4}\qquad\mathrm{(B)}\ \frac{59}{4}\qquad\mathrm{(C)}\ \frac{61}{4}\qquad\mathrm{(D)}\ \frac{63}{4}\qquad\mathrm{(E)}\ \frac{65}{4}</math> | <math>\mathrm{(A)}\ \frac{57}{4}\qquad\mathrm{(B)}\ \frac{59}{4}\qquad\mathrm{(C)}\ \frac{61}{4}\qquad\mathrm{(D)}\ \frac{63}{4}\qquad\mathrm{(E)}\ \frac{65}{4}</math> | ||
− | + | == Solution == | |
− | ==Solution== | ||
=== Solution 1 === | === Solution 1 === | ||
Let the legs of the triangle have lengths <math>a,b</math>. Then, by the [[Pythagorean Theorem]], the length of the hypotenuse is <math>\sqrt{a^2+b^2}</math>, and the area of the triangle is <math>\frac 12 ab</math>. So we have the two equations | Let the legs of the triangle have lengths <math>a,b</math>. Then, by the [[Pythagorean Theorem]], the length of the hypotenuse is <math>\sqrt{a^2+b^2}</math>, and the area of the triangle is <math>\frac 12 ab</math>. So we have the two equations | ||
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=== Solution 4 === | === Solution 4 === | ||
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Let <math>a</math> and <math>b</math> be the legs of the triangle, and <math>c</math> the hypotenuse. | Let <math>a</math> and <math>b</math> be the legs of the triangle, and <math>c</math> the hypotenuse. | ||
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The answer is choice (B). | The answer is choice (B). | ||
− | ===Solution 5=== | + | === Solution 5 === |
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Let <math>a</math>, <math>b</math>, and <math>c</math> be the sides of the triangle, with <math>c</math> as the hypotenuse. | Let <math>a</math>, <math>b</math>, and <math>c</math> be the sides of the triangle, with <math>c</math> as the hypotenuse. | ||
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Finally, subtracting this from our original value of 32, we get <math>\frac{59}{4}</math>, or <math>B</math>. | Finally, subtracting this from our original value of 32, we get <math>\frac{59}{4}</math>, or <math>B</math>. | ||
− | ==See | + | == See Also == |
{{AMC10 box|year=2008|ab=A|num-b=17|num-a=19}} | {{AMC10 box|year=2008|ab=A|num-b=17|num-a=19}} | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 23:32, 18 October 2020
Contents
[hide]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
Re-arranging the first equation and squaring,
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
![\begin{align*} a+b+c &= 32 \\ 2ab &= 80. \\ \end{align*}](http://latex.artofproblemsolving.com/9/b/1/9b1ad9609b79bdcfc10cd8052a111b659e5b0f13.png)
Subtracting from both sides of the first equation and squaring both sides, we get
![\begin{align*} (a+b)^2 &= (32 - c)^2\\ a^2 + b^2 + 2ab &= 32^2 + c^2 - 64c.\\ \end{align*}](http://latex.artofproblemsolving.com/8/f/1/8f13a9ea7ec9417ab1fab36f5ae1d9ecb5ef4179.png)
Now we substitute in as well as
into the equation to get
![\begin{align*} 80 &= 1024 - 64c\\ c &= \frac{944}{64}. \end{align*}](http://latex.artofproblemsolving.com/a/a/1/aa184abe19e53f4bb2e8712ad5b4e6a2e0aaf82d.png)
Further simplification yields the result of .
Solution 4
Let and
be the legs of the triangle, and
the hypotenuse.
Since the area is 20, we have .
Since the perimeter is 32, we have .
The Pythagorean Theorem gives .
This gives us three equations with three variables:
![$ab = 40 \\ a + b + c = 32 \\ c^2 = a^2 + b^2$](http://latex.artofproblemsolving.com/a/b/3/ab34477dc4e0b22ad043a9d2ae5228ade3611d17.png)
Rewrite equation 3 as .
Substitute in equations 1 and 2 to get
.
![$c^2 = (32-c)^2 - 80 \\\\ c^2 = 1024 - 64c + c^2 - 80 \\\\ 64c = 944 \\\\ c = \frac{944}{64} = \frac{236}{16} = \frac{59}{4}$](http://latex.artofproblemsolving.com/b/b/0/bb0955d038919472bbefb6e672b0b6c7192fb561.png)
The answer is choice (B).
Solution 5
Let ,
, and
be the sides of the triangle, with
as the hypotenuse.
We know that .
According to the Pythagorean Theorem, we have .
We also know that = 40, since the area of the triangle is 20.
We substitute into
to get
.
Moving the to the left, we again rewrite to get
.
We substitute our value of 32 for twice into our equation and subtract to get
.
Finally, subtracting this from our original value of 32, we get , or
.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.