Which ratio represents the lengths of the sides of a 45-45-90 triangle?

In a 45-45-90 triangle, the angles are 45 degrees, 45 degrees, and 90 degrees, indicating that it is an isosceles right triangle. The sides opposite the 45-degree angles are of equal length, and those lengths can be represented by the ratio of 1:1. The side opposite the 90-degree angle, which is the hypotenuse, can be derived using the properties of right triangles.

In a 45-45-90 triangle, the lengths of the sides can be determined using the relationship that the length of the hypotenuse is equal to the length of a leg multiplied by the square root of 2. Thus, if the lengths of the legs are both 1, the hypotenuse will be (1 \cdot \sqrt{2}). Therefore, the complete ratio of the sides becomes 1:1:√2, which accurately represents the lengths of the sides of a 45-45-90 triangle.

This means that the correct answer showcases how the two legs are equal and the hypotenuse is related to those legs through multiplication by ( \sqrt{2} ). Other options presented either incorrectly represent the side lengths or do not