2005 AMC 8 Problems/Problem 15

Problem

How many different isosceles triangles have integer side lengths and perimeter 23?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$

Solution

Let $b$ be the base of the isosceles triangles, and let $a$ be the lengths of the other legs. From this, $2a+b=23$ and $b=23-2a$ . From triangle inequality, $2a>b$ , then plug in the value from the previous equation to get $2a>23-2a$ or $a>5.75$ . The maximum value of $a$ occurs when $b=1$ , in which from the first equation $a=11$ . Thus, $a$ can have integer side lengths from $6$ to $11$ , and there are $\boxed{\textbf{(C)}\ 6}$ triangles.

2005 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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
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2005 AMC 8 Problems/Problem 15

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