Understanding the Side Ratio of a 45-45-90 Triangle: Explained and Demonstrated

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, each of the two acute angles is 45 degrees, making the triangle an isosceles right triangle. This means that the two legs (sides adjacent to the 45-degree angles) are congruent, and the length of the hypotenuse (side opposite the right angle) is equal to the length of the legs times the square root of 2 (√2).

To understand this side ratio, let’s assume the length of each leg is ‘x’. Since it is an isosceles triangle, both legs are of equal length. Therefore, the side lengths can be expressed as x:x:x√2.

To find the length of the hypotenuse, we use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. By applying this theorem, we get:

x^2 + x^2 = (x√2)^2
2x^2 = 2x^2

This implies that the Pythagorean theorem is satisfied, confirming that the side lengths of the triangle are consistent with a 45-45-90 triangle.

So, in summary, the side ratio of a 45-45-90 triangle is 1:1:√2, with the two legs being congruent and the length of the hypotenuse being equal to the length of the legs times the square root of 2.

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