What are the side ratios of a 45-45-90 special right triangle?

In a 45-45-90 triangle, the two legs are of equal length, which we can denote as ( x ). The hypotenuse, according to the properties of this special right triangle, is given by the formula ( x\sqrt{2} ). This relationship arises from the Pythagorean theorem, where the length of the hypotenuse is the square root of the sum of the squares of the other two sides. Since both legs are equal, the calculation simplifies to ( \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2} ).

Thus, the side ratios can be expressed as ( x, x, x\sqrt{2} ), which perfectly matches the choice indicating these side ratios. This unique characteristic of 45-45-90 triangles makes them predictable and easy to work with in geometric problems, reinforcing why it is essential to know these specific ratios.