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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.

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

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.

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.