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Here is the problem: A right triangle has legs 5 and 12. Find the hypotenuse.

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Find the hypotenuse of a right triangle from its two legs.

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Write the Pythagorean theorem. The hypotenuse c is the side opposite the right angle and is always the longest side.

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Substitute the legs and square them.

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Take the positive square root. Lengths are positive, so only the positive root is used.

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Area and perimeter.

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Acute angles. Angle A is opposite leg a and angle B is opposite leg b; the right angle C is 90°.

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So the answer is c equals 13; area equals 30; perimeter approximately 30; angles approximately 22.619865°, 67.380135°, 90°.

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In every right triangle the square on the hypotenuse equals the sum of the squares on the two legs. Knowing any two sides fixes the third, and the angles follow from the tangent ratio.

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Check: a squared plus b squared equals 169 and c squared equals 169, which agree.

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(5, 12, 13) is a Pythagorean triple: all three sides are whole numbers.

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Angle sum: 90° plus 22.619865° plus 67.380135° equals 180°.