If you have a 45-45-90 triangle, what is the ratio of the lengths of the sides?

In a 45-45-90 triangle, which is an isosceles right triangle, the lengths of the sides are in a specific ratio derived from the properties of the triangle. In such a triangle, both legs are of equal length, commonly referred to as '1' for simplicity. The hypotenuse can be calculated using the Pythagorean theorem, which in this case states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Since both legs are of equal length (let's call them 'x'), the equation becomes:

[

hypotenuse^2 = x^2 + x^2 = 2x^2

]

Taking the square root gives the hypotenuse as:

[

hypotenuse = \sqrt{2}x

]

Substituting '1' for 'x' (as a generic representation for the length of the legs), the hypotenuse is then (\sqrt{2}).

Therefore, the ratio of the lengths of the sides in a 45-45-90 triangle is 1:1 for the legs and (\sqrt{2}) for the hypotenuse, resulting in the