### Instruction: If the lengths of two sides of a right triangle are 5 and 12 units, what is the least possible length, in units, of the third side? Express your answer in simplest radical form.### Response: I know that the Pythagorean theorem relates the lengths of the sides of a right triangle by the equation a^2 + b^2 = c^2, where c is the hypotenuse and a and b are the legs.
Since I don't know which side is the hypotenuse, I'll try both possibilities and see which one gives me a smaller value for the third side.
If I assume that the hypotenuse is 12, then the other leg must satisfy 5^2 + b^2 = 12^2, or b^2 = 144 - 25 = 119.
Taking the square root of both sides, I get b = sqrt(119), which is already in simplest radical form.
If I assume that the hypotenuse is the unknown side, then it must satisfy 5^2 + 12^2 = c^2, or c^2 = 25 + 144 = 169.
Taking the square root of both sides, I get c = sqrt(169) = 13.
Comparing the two values, I see that sqrt(119) is smaller than 13, since 119 is smaller than 169.