**Related Pages**

Pythagorean Triples

Types Of Triangles

Area Of Triangles

More Geometry Lessons

### Recognizing Special Right Triangles

Recognizing special right triangles can provide ashortcut when answering some geometry questions. A special right triangle is a right trianglewhose sides are in a particular ratio, called the Pythagorean Triples. Youcan also use the Pythagorean theorem", but if you can see thatit 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, Pythagoreantriples, videos and worksheets.

#### What is a 45°-45°-90° Triangle?

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 bea 45°-45°-90° triangle.

You can also recognize a 45°-45°-90° triangle by the angles. A right trianglewith 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 ofthe 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 ratiox: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 hypotenuseis 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. Ifthe third value of the ratio n:n:n√2 is 4√2 then the lengths ofthe other two sides must 4.

Answer: Thelengths of the two sides are both 4 inches.

#### What is a 30°-60°-90° Triangle?

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 asyou 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. Aright triangle with a 30° angle or 60° angle must be a30°-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 ofthe 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: Thelength of the hypotenuse is 8 inches.

**Example 2:**

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

**Solution:**

Step 1:This is a right triangle with a 30° angleso 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 twosides 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.

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**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.

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**How to Solve Special Right Triangles?**

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

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**Special Right Triangles in Geometry**

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

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

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**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 ratioof 3:4:5. When you are given the lengths of two sides of a righttriangle, check the ratio of the lengths to see if it fits the3:4:5 ratio.

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

**Example 1: **

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

**Solution:**

Step 1: Test the ratio of thelengths 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 thehypotenuse 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 itfits 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 arein the ratio of 5:12:13. It is another example of a special righttriangle.

**Example:**

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

**Concepts and patterns of Pythagorean triples**

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**Examples and families of Pythagorean Triples**

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