Geometry: Special Right Triangles

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 thie lesson, 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

3–4–5 Triangles

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.

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 a² + b² = c². Two other commonly used Pythagorean Triples are (8, 15, 17) and (7, 24, 25)

45º-45º-90º Triangles

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 .

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

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  is 3 then the length of the third side is

Answer: The length of the hypotenuse is  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° then it must be a 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 inches and one of the angles is 45°.

Solution:

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

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

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

30º-60º-90º Triangles

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

Solution:

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

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

Step 3:  Calculate the third side.

2n = 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.

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 , we get that 2n = 8 Þ n = 4.

Substituting n = 4 into the first and second value of the ratio we get that the other two sides are 4 and .

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

Special Triangles - Important Angles - 30, 45, 60

Solving 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.

Pythagorean Triples

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