# Special Right Triangles Lesson

Special Right Triangles are specific types of right triangles with special properties relating their angles and side lengths. The two most common types of special right triangles are the 45-45-90 and 30-60-90 triangles.

In the **45-45-90 Special Right Triangle** the two acute angles are congruent, each measuring 45°. The sides opposite of those acute angles are also congruent. If we state that the two legs that are opposite the 45° angles are x, then using the Pythagorean Theorem, we conclude that the length of the hypotenuse in a 45-45-90 triangle is x√2. The ratio of the legs to the hypotenuse is Leg:Leg:Hypotenuse 1:1:√2.

In the **30-60-90 Special Right Triangle** the two acute angles are 30° and 60°. The side opposite the 30° angle is x, the side opposite of the 60° angle is x√3 and using the Pythagorean Theorem, we can conclude that the hypotenuse in the 30-60-90 triangle is 2x. The ratio of the legs to the hypotenuse is

Short Leg: Long Leg : Hypotenuse: 1:√3:2.

We use the ratios in terms of x to solve for missing side lengths in these two special right triangles. For instance, if I knew that the hypotenuse of a 30-60-90 triangle was 10, then based on the 30-60-90 Special Right Triangle ratios, that the value of x, and the shorter leg is 5, and therefore, the longer leg would be 5√3.

Examples

**Example 1: **Find the shorter leg and the hypotenuse of a 30-60-90 triangle if the longer leg is 4√3 units.

**Solution:** Since this is a 30-60-90 triangle we know that the ratios of the short leg: long leg: hypotenuse is 1:√3:2, in terms of x, that ratio is x:x√3:2x. Since we are given the long leg is 4√3 and we know the ratio of the long leg is x√3.Therefore we can set those expressions equal to each other to solve for x.

4√3 = x√3

Which means that x = 4,and hence the short leg is 4 units. We can now substitute 4 into the ratio for the hypotenuse, , which means that the hypotenuse is 8 units.

**Example 2:** Find the length of the legs in a 45-45-90 triangle if the hypotenuse is 8√2 units.

**Solution:** Since this is a 45-45-90 triangle we know that the ratios of each leg to the hypotenuse is 1:1:√2. Written in terms of x, means that the ratio leg: leg: hypotenuse is x:x:x√2. Since we know that the length of the hypotenuse is 8√2, using the ratio x√2 of the hypotenuse, means that x = 8.

Therefore, the length of each leg is 8 units.

**Example 3:** Find the length of the short leg and long leg in a 30-60-90 triangle if the hypotenuse is 10 units.

**Solution: **Since this is a 30-60-90 triangle we know that the ratios of the short leg: long leg: hypotenuse is 1:√3:2, in terms of x, that ratio is x:x√3:2x. Since we know the length of the hypotenuse is 10 units, use the ratio 2x, we can set 10 = 2x and solve for x by dividing each side by 2, which results in x = 5.

Since we know that x = 5,we know that the short leg is 5 units, and the long leg is 5√3 units.

FAQs on Special Right Triangles

**1) What are special right triangles?**

Special right triangles are right triangles that have angles with specific measurements (such as 45 degrees, 30 degrees, and 60 degrees) and side lengths that follow certain ratios. The two main types are the 45-45-90 triangle and the 30-60-90 triangle.

**2) What is a 45-45-90 triangle?**

A 45-45-90 triangle is a right triangle where both non-right angles are 45 degrees. The sides are in the ratio 1:1:√2

**3) What is a 30-60-90 triangle?**

A 30-60-90 triangle is a right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides are in the ratio 1:√3:2

**4) Why are they called special right triangles?**

They are called special right triangles because they have properties that make them particularly easy to work with in geometry and trigonometry. Their angles and side ratios follow simple patterns, which simplifies calculations.

**5) How do you find the side lengths of special right triangles?**

For a 45-45-90 triangle, if one leg has a length of x, then the other leg also has a length of x, and the hypotenuse has a length of x√2. For a 30-60-90 triangle, if the shorter leg has a length of x, then the longer leg has a length of x√3, and the hypotenuse has a length of 2x.

**6) What are the applications of special right triangles?**

Special right triangles are commonly used in geometry, trigonometry, and physics to solve problems involving right triangles. They simplify calculations for finding side lengths, angles, areas, and perimeters.

**7) Are there any other types of special right triangles?**

No, the45-45-90 triangle and the 30-60-90 triangle are the only two types of special right triangles. However, other triangles may have special properties, but they don’t follow the same simple patterns as these two types.