Mastering Special Right Triangles: Understanding the Ratios and Applications of 45-45-90 and 30-60-90 Triangles

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Special right triangles are two types of right triangles that have certain angles and side ratios that remain constant regardless of the size of the triangle. These two types of special right triangles are the 45-45-90 triangle and the 30-60-90 triangle.

The 45-45-90 triangle is an isosceles right triangle where each of the two acute angles are 45 degrees. Therefore, the ratio of the sides of this triangle is one unit of the hypothenuse to square root of 2 on each of the legs. This can be represented as:

leg = (1/√2) x hypothenuse
hypothenuse = √2 x leg

The 30-60-90 triangle is a right triangle whose angles measure 30 degrees, 60 degrees and 90 degrees. The shorter leg opposite the 30 degree angle is half the length of the hypothenuse. The longer leg, opposite the 60-degree angle, is √3 times the length of the shorter leg. The ratio of the sides of this triangle can be represented as:

short leg = (1/2) x hypothenuse
long leg = √3/2 x short leg
hypothenuse = 2 x short leg

Knowing these ratios help in solving geometric and trigonometric problems involving these special right triangles. For example, if the length of one leg of a 30-60-90 triangle is 5 inches, then the length of the hypothenuse would be 10 inches and the length of the longer leg would be 5√3 inches. These ratios can also be used to simplify trigonometric functions and express them in exact values instead of approximations.

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