The Side Ratio of a 45-45-90 Triangle: Understanding 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 legs (the sides opposite the 45-degree angles) have the same length, and the length of the hypotenuse (the side opposite the 90-degree angle) is √2 times the length of the legs.

To understand why this ratio holds true, let’s consider the properties of a 45-45-90 triangle.

In a unit square (a square with side length 1), if we draw the diagonal from one corner to the opposite corner, we get a 45-45-90 triangle.

Let’s label the side length of the square as “a”. Since the square is a 1×1 square, we can say that a = 1.

Now, let’s use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we have:
hypotenuse^2 = leg^2 + leg^2
hypotenuse^2 = 1^2 + 1^2
hypotenuse^2 = 1 + 1
hypotenuse^2 = 2
hypotenuse = √2

So, in the unit square, the length of the hypotenuse is √2.

Since the 45-45-90 triangle is simply a scaled version of the unit square, we can conclude that in any 45-45-90 triangle, the ratio of the lengths of the legs to the length of the hypotenuse is 1:1:√2.

Therefore, the side ratio of a 45-45-90 triangle is 1:1:√2.

More Answers: