Understanding the Side Ratio of a 45-45-90 Triangle: Exploring the 1:1:√2 Relationship

The side ratio of a 45-45-90 triangle is …

The side ratio of a 45-45-90 triangle is 1:1:√2

The side ratio of a 45-45-90 triangle is 1:1:√2.

In a 45-45-90 triangle, the two acute angles are both 45 degrees, making it an isosceles triangle. This means that the two equal sides are opposite to the 45-degree angles.

To determine the side ratio, let’s consider the lengths of the sides. Assume that the two equal sides have a length of “x”, and the hypotenuse has a length of “h”.

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, we have:

x^2 + x^2 = h^2 (since the two equal sides are x)

By simplifying the equation, we get:

2x^2 = h^2

To find the relationship between the sides, we need to rearrange the equation to isolate “x”:

h^2 = 2x^2

Taking the square root of both sides:

√(h^2) = √(2x^2)

h = x√2

This shows that the hypotenuse is √2 times longer than the equal sides. In terms of side ratio, it can be expressed as 1:1:√2, where the two equal sides have a ratio of 1 and the hypotenuse has a ratio of √2.

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