Final answer:
The length of the hypotenuse in a right triangle with one angle of 45° and a side opposite that angle measuring 'x' meters, is x√2 meters, determined by the Pythagorean theorem.
Step-by-step explanation:
To find the length of the hypotenuse in a right triangle with an angle θ of 45° and the side opposite to θ known, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b) or c² = a² + b². Since the angle is 45°, we have an isosceles right triangle, meaning both the legs (a and b) are of equal length. To solve for the hypotenuse (c), we use the formula c = √(a² + b²). If the side opposite to θ is 'x' meters long, then the hypotenuse is c = √x² + x² = √2x² = x√2. Therefore, if the length of the side opposite is 'x' meters, the hypotenuse measures approximately x√2 meters.