# Right Triangles Demystified

## What is a Right Triangle?

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A right triangle is a special type of triangle in which one of the angles measures When we do have a right triangle, the measure of the remaining two angles that are not the right angle **CANNOT** be equal to , as the sum of all interior angles of a triangle must be equal to . A angle is usually notated by the ∟ symbol. Here are some examples:

**Figure #1**

**Figure #2**

**Figure #3**

In each of these triangles, the angle is located where the ∟ symbol is.

**BONUS TIP****:**

In a right triangle, the longest side is known as the hypotenuse. The hypotenuse will **ALWAYS** be the side opposite of the angle. (Since the largest angle in a Right Triangle is always and the longest side is always opposite the largest interior angle). The two shorter sides are known as legs.

## What is a special right triangle?

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Now that we know about right triangles, let’s talk about two special types of right triangles. Each triangle will have either of the following measures for its three angles.

- - **OR ** - -

### 30° – 60° – 90°

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In a - - right triangle, one angle measures , one angle measures , and one angle measures . The sides of the triangle opposite to each angle have a ratio of , respectively.

**Figure #4**

### 45° – 45° – 90°

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In a - - right triangle, one angle measures and two angles measure . The sides of the triangle opposite to each angle have a ratio of .

**Figure #5**

Using these properties, we can easily solve for the remaining sides of a right triangle if given only one side measurement and knowing that we have a special right triangle.

## Fun facts about right triangles:

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## Examples

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**Find the missing side,****, of each given triangle.**

**#1**

For the problem in **Figure #6**, we are only given one side of the triangle. To solve for the missing side, , let’s first solve for the remaining angle. We can do that by setting up the following equation:

Where is the value of the missing angle.

Simplifying we get

Solving for we get

Since , we can see that we have a - - triangle. Using this information, we can solve for the missing side .

The side we are given is opposite of . The side opposite of is equal to . Therefore,

Solving for we get

Rationalizing we get

The side is equal to . So,

**#2**

In **Figure #7**, we will once again begin with solving for the missing angle. We’ll call it . So,

Solving for we get

Since , we can see that we have a - - triangle. Using this information, we can solve for the missing side .

The side we are given is opposite of . The side opposite of is equal to . Therefore,

Solving for we get

Rationalizing, we get

Simplifying, we get

The side is equal to . So,

Author: Mr. Vernon Sullivan, is a tutor at FPLA, a premier 1-on-1 tutoring center HQ in Miami FL. He teaches Algebra, Geometry, Pre-Cal, ACT, SAT, SSAT, HSPT, PERT, ASVAB and other test prep programs.

Mrs. Emimmal Sekar Proofread this article. Mr. Arikaran Kumar manages the website and the social media outreach.