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A right triangle has side lengths 6 centimeters and 8 centimeters. Is it possible for there to be more than one right triangle with whole-number side lengths that includes the given side lengths? Explain.

User Tommo
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The given side lengths, 6 cm and 8 cm, satisfy the Pythagorean theorem because $6^2 + 8^2 = 100 = 10^2$, which means that the triangle is a right triangle.

To determine if there is more than one right triangle with whole-number side lengths that includes these given side lengths, we need to check if there exist other pairs of whole numbers that satisfy the Pythagorean theorem and include 6 cm and 8 cm as two of their sides.

Let's call the third side of the right triangle x. Then we have:

62+82=x26^2 + 8^2 = x^262+82=x2

which simplifies to:

36+64=x236 + 64 = x^236+64=x2

100=x2100 = x^2100=x2

x=100=10x = \sqrt{100} = 10x=100

​=10

So the only other possible triangle with whole-number side lengths that includes the given side lengths has side lengths of 6 cm, 8 cm, and 10 cm.

Therefore, there is only one other right triangle with whole-number side lengths that includes the given side lengths, and it is unique.

User Kmdent
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