Final answer:
Using the Pythagorean theorem, the shortest possible length for the third side of a right triangle with sides of 26 and 10 units is 24 units, and the longest possible length is approximately 27.86 units.
Step-by-step explanation:
The question involves finding the possible lengths for the third side of a right triangle when two sides are given. We have one side of 26 units and another of 10 units. Using the Pythagorean theorem, which states that in a right triangle the sum of the squares of the two legs (a and b) is equal to the square of the hypotenuse (c), we have:
If the side with 26 units is the hypotenuse (the longest side), the other side will be shorter than the 10-unit side. We calculate the shorter length (x) using the formula x = √(c² - b²) = √(26² - 10²) = √(676 - 100) = √576 = 24 units.
If the side with 26 units is not the hypotenuse, then the hypotenuse (y) is calculated using the formula y = √(a² + b²) = √(10² + 26²) = √(100 + 676) = √776 ≈ 27.86 units.
The shortest possible side length is 24 units and the longest possible side length for the third side of the right triangle is about 27.86 units.