A special right triangle is a right triangle with some special feature that makes the calculations easier. There are two types of special right triangles:
30° – 60° – 90°
This is a right triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30°, 60°, and 90°. The sides are in the ratio x : x√3 : 2x or 1 : √3 : 2.
The side opposite 30° is the shortest side, x or 1. The side opposite 60° is x√3 or √3. The side opposite 90° is the longest side (hypotenuse), 2x or 2.
Example: If the shortest side of 30° – 60° – 90° right triangle is 5, what is the measure of the other two sides?
Given x = 5.
The other two sides are 2x = 10 which is the longest side (hypotenuse) and x√3 = 5√3.
45° – 45° – 90°
This is a special right triangle whose three angles are in 1 : 1 : 2 and measure 45°,45° and 90°. The sides of this triangle are in ratio 1 : 1 : √2 or x : x : x√2. This triangle forms an isosceles right triangle.
Example: If one of the equal sides of a 45° – 45° – 90° right triangle is 6, what are the measures of the other two sides?
Given x = 6,
The other two sides are x = 6 and x√2 = 6√2.
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