In these lessons, we will learn to

- about special right triangles: 45°-45°-90° triangles and 30°-60°-90° triangles.
- how to solve special right triangles.
- some examples of Pythagorean Triples: 3-4-5 triangles, 5-12-13 triangles.

**Related Pages**

Pythagorean Triples

Types Of Triangles

Area Of Triangles

More Geometry Lessons

Recognizing special right triangles can provide a shortcut when answering some geometry questions. A special right triangle is a right triangle whose sides are in a particular ratio, called the Pythagorean Triples. You can also use the Pythagorean theorem", but if you can see that it is a special triangle it can save you some calculations.

The following figures show some examples of special right triangles and Pythagorean Triples. Scroll down the page if you need more explanations about special right triangles, Pythagorean triples, videos and worksheets.

A 45°-45°-90° triangle is a special right triangle whose angles are 45°, 45° and 90°. The lengths of the sides of a 45°-45°-90° triangle are in the ratio of 1 : 1 : √2.

A right triangle with two sides of equal lengths must be a 45°-45°-90° triangle.

You can also recognize a 45°-45°-90° triangle by the angles. A right triangle with a 45° angle must be a 45°-45°-90° special right triangle.

Side1 : Side2 : Hypotenuse = x : x : x√2

**Example 1:**

Find the length of the hypotenuse of a right triangle if the lengths of
the other two sides are both 3 inches.

**Solution:**

Step 1:
This is a right triangle with two equal sides so it must be a 45°-45°-90°
triangle.

Step 2: You are given that the both the sides are 3. If the first and second value of the ratio x:x:x√2 is 3 then the length of the third side is 3√2.

Answer: The length of the hypotenuse is 3√2 inches.

**Example 2:**

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse
is 4√2 inches and one of the angles is 45°.

**Solution:**

Step 1:
This is a right triangle with a 45°-45°-90° triangle.

You are given that the hypotenuse is 4√2. If the third value of the ratio n:n:n√2 is 4√2 then the lengths of the other two sides must 4.

Answer: The lengths of the two sides are both 4 inches.

Another type of special right triangles is the 30°-60°-90° triangle. This is right triangle whose angles are 30°-60°-90°. The lengths of the sides of a 30°-60°-90° triangle are in the ratio of 1 : √3 : 2.

You can also recognize a 30°-60°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°-60°-90° special right triangle. A right triangle with a 30° angle or 60° angle must be a 30°-60°-90° special right triangle.

Side1 : Side2 : Hypotenuse = x : x√3 : 2x

**Example 1:**

Find the length of the hypotenuse of a right triangle if the lengths of
the other two sides are 4 inches and 4&dadic;3 inches.

**Solution:**

Step 1:
Test the ratio of the lengths to see if it fits the n:n√2:2n ratio.

4:4√3:? = x:x√3:2x

Step 2:
Yes, it is a 30°-60°-90° triangle for *x* = 4

Step 3: Calculate the third side.

2*x* = 2 × 4 = 8

Answer: The
length of the hypotenuse is 8 inches.

**Example 2:**

Find the lengths of the other two sides of a right triangle if the
length of the hypotenuse is 8 inches and one of the angles is 30°.

**Solution:**

Step 1:
This is a right triangle with a 30° angle
so it must be a 30°-60°-90° triangle.

You are given that the hypotenuse is 8.

Substituting 8 into the third value of the ratio x:x√3:2x, we get that 2*x* = 8 ⇒ *x* = 4.

Substituting *x* = 4 into the first and second value of the ratio we get that the other two
sides are 4 and 4√3.

Answer: The lengths of the two sides are 4 inches and 4√3 inches.

**Special Triangles - Important Angles - 30°, 45°, 60°**

45°-45°-90° Triangles, 30°-60°-90° Triangles.

**Basic triangles you should know**

The triangles are classified by side and by angle.

In this video you will learn:

- 3-4-5 triangles and similar triangles.
- 5-12-13 triangles and similar triangles.
- 45-45-90 right triangles.
- 30-60-90 triangles.
- equilateral triangles.
- relationship between equilateral and 30-60-90 triangles.

**How to Solve Special Right Triangles?**

When solving special right triangles,
remember that a 30-60-90 triangle has a hypotenuse twice as long
as one of the sides, and a 45-45-90 triangle has two equal sides.

**Special Right Triangles in Geometry**

45-45-90 and 30-60-90 degree triangles.

Discuss two special right triangles, how to derive the formulas to find the lengths of the sides of the triangles by knowing the length of one side, and a few examples using them.

**What are Pythagorean Triples?**

Any group of 3 integer values that satisfies the equation:
a^{2} + b^{2} = c^{2} is called a Pythagorean Triple.
Any triangle that has sides that form a Pythagorean Triple must be a right triangle.
Some examples of Pythagorean Triple triangles are: 3-4-5 Triangles and 5-12-13 Triangles.

**What is a 3-4-5 Triangle?**

A 3-4-5 triangle is right triangle whose lengths are in the ratio
of 3:4:5. When you are given the lengths of two sides of a right
triangle, check the ratio of the lengths to see if it fits the
3:4:5 ratio.

Side1 : Side2 : Hypotenuse = 3n : 4n : 5n

**Example 1: **

Find the length of the hypotenuse of a
right triangle if the lengths of the other two sides are 6 inches and 8 inches.

**Solution:**

Step 1: Test the ratio of the
lengths to see if it fits the 3n : 4n : 5n ratio.

6 : 8 : ? = 3(2) : 4(2) : ?

Step 2: Yes, it is a 3-4-5 triangle for n = 2.

Step 3: Calculate the third side.

5n = 5 × 2 = 10

Answer: The length of the hypotenuse is 10 inches.

**Example 2: **

Find the length of one side of a right triangle if the length of the
hypotenuse is 15 inches and the length of the other side is 12 inches.

**Solution:**

Step 1: Test the ratio of the lengths to see if it
fits the 3n : 4n : 5n ratio.

? : 12 : 15 = ? : 4(3) : 5(3)

Step 2: Yes, it is a 3-4-5 triangle for n = 3.

Step 3: Calculate the third side.

3n = 3 × 3 = 9

Answer: The length of the side is 9 inches.

**What is a 5-12-13 Triangle?**

A 5-12-13 triangle is a right-angled triangle whose lengths are
in the ratio of 5:12:13. It is another example of a special right
triangle.

**Example:**

3-4-5 and 5-12-13 are examples of the Pythagorean
Triple. They are usually written as (3, 4, 5) and (5, 12,
13). In general, a Pythagorean triple consists of three
positive integers such that a^{2} + b^{2} = c^{2}. Two
other commonly used Pythagorean Triples are (8, 15, 17) and (7,
24, 25)

**Concepts and patterns of Pythagorean triples**

**Examples and families of Pythagorean Triples**

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