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 equal, measuring 45 degrees each. Since the sum of the angles in any triangle is 180 degrees, the remaining angle (opposite the hypotenuse) must be 180 – 2(45) = 90 degrees.
Let’s label the sides of the triangle. The two legs (the two sides opposite the 45-degree angles) will have the same length, so we can call them x and x. The hypotenuse (the side opposite the 90-degree angle) can be denoted as c.
By the Pythagorean theorem, we know that the sum of the squares of the lengths of the two legs equals the square of the hypotenuse. So, we have:
x^2 + x^2 = c^2
Simplifying this equation, we get:
2x^2 = c^2
To find the values of the side lengths, we need to determine the ratio between them. Dividing both sides of the equation by x^2, we have:
2 = (c/x)^2
Taking the square root of both sides, we get:
√2 = c/x
Therefore, the ratio of the hypotenuse to one of the legs in a 45-45-90 triangle is √2:1.
You can also multiply each side ratio by a common factor, which will still maintain the same proportion. So, for example, the side ratio could be 2√2:2, or 3√2:3, and so on.
I hope this explanation helps! Let me know if you have any further questions.
More Answers:
Understanding the Pythagorean Theorem: Identifying right triangles and their characteristicsHow to Determine if a Triangle is Acute: The Side Length Approach and Triangle Inequality Theorem
Determining If a Triangle is Obtuse: Using Side Lengths and the Pythagorean Theorem