# Right Triangles Demystified

Last updated on J by Arikaran Kumar

Right Triangles

## What is a Right Triangle?

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?

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°

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.

• The side opposite to is equal to
• The side opposite to is equal to
• The side opposite to is equal to

Figure #4

### 45° – 45° – 90°

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

• The sides opposite to each angle is equal to
• The side opposite to is equal to

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:

• Right triangle has many synonyms. Right-angled triangle, orthogonal triangle, rectangled triangle; the common thread, a triangle having one of the angles measuring .
• The foundations of trigonometry can be traced to the study of the relationship between the sides and the angles, other than the , of a right triangle.

## Examples

Find the missing side,, of each given triangle.

#1

Figure #6

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

Figure #7

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.

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