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2008 AMC 10B Problems/Problem 15

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Problem

How many right triangles have integer leg lengths a and b and a hypotenuse of length b+1, where b<100?

\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 7\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 10

Solution

By the pytahagorean theorem, a^2+b^2=b^2+2b+1

This means that a^2=2b+1.

We know that a,b>0, and that b<100.

We also know that a must be odd, since the right

side is odd.

So a=3,5,7,9,11,13, and the answer is \boxed{A}.

See also

2008 AMC 10B (Problems • 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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