To find the length of the third side of a right triangle given two sides, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).
Given:
- One side (leg) = 5
- The other side (hypotenuse) = 13
- The third side (let's say, b) is the other leg we need to find.
According to the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs of the triangle and \( c \) is the hypotenuse.
Plugging in the known values: \[ 5^2 + b^2 = 13^2 \]
Calculating the squares: \[ 25 + b^2 = 169 \]
Now, isolate \( b^2 \): \[ b^2 = 169 - 25 \] \[ b^2 = 144 \]
Taking the square root of both sides gives: \[ b = \sqrt{144} \] \[ b = 12 \]
Therefore, the length of the third side is: \[ \boxed{12} \]
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