Difference between revisions of "2007 AMC 12A Problems/Problem 10"

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Since 3-4-5 is a [[Pythagorean triple]], the triangle is a [[right triangle]]. Since the hypotenuse is a [[diameter]] of the [[circumcircle]], the hypotenuse is <math>2r = 6</math>. Then the other legs are <math>\frac{24}5=4.8</math> and <math>\frac{18}5=3.6</math>. The area is <math>\frac{4.8 \cdot 3.6}2 = 8.64\ \mathrm{(A)}</math>
 
Since 3-4-5 is a [[Pythagorean triple]], the triangle is a [[right triangle]]. Since the hypotenuse is a [[diameter]] of the [[circumcircle]], the hypotenuse is <math>2r = 6</math>. Then the other legs are <math>\frac{24}5=4.8</math> and <math>\frac{18}5=3.6</math>. The area is <math>\frac{4.8 \cdot 3.6}2 = 8.64\ \mathrm{(A)}</math>
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==Solution 2==
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The hypotenuse of the triangle is a diameter of the circumcircle, so it has length <math>2 \cdot 3 = 6</math>. The triangle is similar to a 3-4-5 triangle with the ratio of their side lengths equal to <math>\frac{6}{5}</math>. The area of a 3-4-5 triangle is <math>\frac{3\cdot 4}{2} = 6</math>.
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The square of the ratio of their side lengths is equal to the ratio of their areas. Call the area of the triangle <math>A</math>. Therefore, <math>\left(\frac{6}{5}\right)^2 = \frac{A}{6} \Longrightarrow \frac{36}{25} = \frac{A}{6} \Longrightarrow A = \frac{36\cdot6}{25} = 8.64\ \mathrm{(A)}</math>
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~mobius247
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==Solution 3 (dimensional analysis)==
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Using the theorem that the hypotenuse of a right triangle inscribed in a circle is the diameter, one can determine that the diameter of the triangle is 6. As a result the ratio between the triangular ratios and their real values is (6/5). Using dimensional analysis, one can see that because the ratio between lengths is (6/5), the ratio between areas is (6/5)^2. This gives ((3 * 4)(6/5)^2)/2 as the area of the triangle or answer choice (A).
  
 
==See also==
 
==See also==

Latest revision as of 03:49, 1 July 2024

The following problem is from both the 2007 AMC 12A #10 and 2007 AMC 10A #14, so both problems redirect to this page.

Problem

A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?

$\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18$

Solution

2007 AMC12A-10.png

Since 3-4-5 is a Pythagorean triple, the triangle is a right triangle. Since the hypotenuse is a diameter of the circumcircle, the hypotenuse is $2r = 6$. Then the other legs are $\frac{24}5=4.8$ and $\frac{18}5=3.6$. The area is $\frac{4.8 \cdot 3.6}2 = 8.64\ \mathrm{(A)}$

Solution 2

The hypotenuse of the triangle is a diameter of the circumcircle, so it has length $2 \cdot 3 = 6$. The triangle is similar to a 3-4-5 triangle with the ratio of their side lengths equal to $\frac{6}{5}$. The area of a 3-4-5 triangle is $\frac{3\cdot 4}{2} = 6$. The square of the ratio of their side lengths is equal to the ratio of their areas. Call the area of the triangle $A$. Therefore, $\left(\frac{6}{5}\right)^2 = \frac{A}{6} \Longrightarrow \frac{36}{25} = \frac{A}{6} \Longrightarrow A = \frac{36\cdot6}{25} = 8.64\ \mathrm{(A)}$

~mobius247

Solution 3 (dimensional analysis)

Using the theorem that the hypotenuse of a right triangle inscribed in a circle is the diameter, one can determine that the diameter of the triangle is 6. As a result the ratio between the triangular ratios and their real values is (6/5). Using dimensional analysis, one can see that because the ratio between lengths is (6/5), the ratio between areas is (6/5)^2. This gives ((3 * 4)(6/5)^2)/2 as the area of the triangle or answer choice (A).

See also

2007 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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
2007 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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

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