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Recognizing special right
triangles in geometry can help you to answer some questions
quicker. A special right triangle is a right triangle whose sides
are in a particular ratio. You can also
use the Pythagorean theorem,
but if you can see that it is a special triangle it can save you
some calculations.

In these lessons, we will learn

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

Related Topics: More Geometry Lessons

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 = 3*n*
: 4*n* : 5*n*

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 3*n *: 4*n *: 5*n*
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

5

n= 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 3*n *: 4*n
*: 5*n*
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

3

n= 3 × 3 = 9

Answer: The length of the side is 9
inches.

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)

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 \(1:1:\sqrt 2 \).

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

Side1 : Side2 : Hypotenuse = \(n:n:n\sqrt 2 \)

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 \(n:n:n\sqrt 2 \) is 3 then the length of the third side is \(3\sqrt 2 \).

Answer: The length of the hypotenuse is \(3\sqrt 2 \) inches.

You can also recognize a 45°-45°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is 45°-45°-90° special right triangle.

A right triangle with a 45° angle must be a 45°-45°-90° special right triangle.

Example 2: Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is \(4\sqrt 2 \) 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 \(4\sqrt 2 \). If the third value of the ratio \(n:n:n\sqrt 2 \) is \(4\sqrt 2 \) then the lengths of the other two sides must 4.

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

Side1 : Side2 : Hypotenuse = \(n:n\sqrt 3 :2n\)

Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and \(4\sqrt 3 \) inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the \(n:n\sqrt 3 :2n\) ratio.

\(4:4\sqrt 3 :? = n:n\sqrt 3 :2n\)

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

Step 3: Calculate the third side.

2

n= 2 × 4 = 8

Answer: The length of the hypotenuse is 8 inches.

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.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 \(n:n\sqrt 3 :2n\), we get that 2*n* = 8 ⇒ *n* = 4.

Substituting *n* = 4 into the
first and second value of the ratio we get that the other two
sides are 4 and \(4\sqrt 3 \).

Answer: The lengths of the two sides are 4 inches and \(4\sqrt 3 \) inches.

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

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

This video outlines the basic triangles you should know. The
triangles are classified by side and by angle.

In this video you will learn: 1) 3-4-5 triangles and similar
triangles 2) 5-12-13 triangles and similar triangles 3) 45-45-90
right triangle and similar triangles 4) 30-60-90 triangle and
similar triangles 5) equilateral triangles 6) relationship between
equilateral and 30-60-90 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.

This video gives some examples and families of Pythagorean Triples