A right triangle has legs 10 and 13. Find the hypotenuse
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We are given a right triangle with legs of length 10 and 13 units. Our goal is to find the length of the hypotenuse. In a right triangle, the hypotenuse is the longest side, opposite the right angle. The two shorter sides are called the legs.
The Pythagorean theorem is a fundamental principle in geometry. It states that in any right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (the legs). This is expressed mathematically as a squared plus b squared equals c squared, where a and b are the legs and c is the hypotenuse.
Now we'll substitute our specific values into the Pythagorean theorem. Our legs have lengths 10 and 13, so we replace a with 10 and b with 13 in the formula. This gives us 10 squared plus 13 squared equals c squared.
Next, we calculate the squares of our leg lengths. Ten squared is 10 times 10, which equals 100. Thirteen squared is 13 times 13, which equals 169. Now our equation becomes 100 plus 169 equals c squared.
Now we add 100 and 169 to get 269. So c squared equals 269. To find c, we take the square root of both sides, giving us c equals the square root of 269. Using a calculator, we find that the square root of 269 is approximately 16.4 units.
Let's verify our answer. We substitute back into the original equation: 10 squared plus 13 squared equals 100 plus 169, which equals 269. The square of the square root of 269 is also 269. Since both sides are equal, our solution is correct. Therefore, the hypotenuse of the right triangle with legs 10 and 13 is the square root of 269, which is approximately 16.4 units.