## Right-angled triangle and the Pythagorean theorem

In this article lets try to develop the Pythagorean theorem by means of the right angle.

So first let’s get some proper ideas about the right-angled triangle.

### What is a right-angled triangle?

If an angle of a triangle is **90**^{o} it is called a right-angled triangle or right triangle.

## Special Right Triangles

### Classification of right triangles according to their sides

### Classification of right triangles according to their Angles

Other than the above two types of right-angled triangles there are two main right-angled triangles that you always search for.

**30 60 90 Triangle**

This is actually a **Scalene Right Angled Triangle**.

**45 45 90 Triangle**

This is an **Isosceles triangle**.

## Terms Related to Right Angled Triangle

The side which is opposite to the right angle and which is the longest side of the triangle is called Hypotenuse.

**BC = HypotenuseAB / AC = BaseAB / AC = Perpendicular**

The sides **AB** and **AC** are also called the sides which include the right angle.

## The Pythagorean Theorem

This is also called as **Pythagoras’ theorem** or **Pythagorean relationship**. This is a fundamental relation in euclidean geometry.

There are several methods to prove the Pythagorean theorem. In this article, I’m going to discuss two methods.

### The Pythagorean Theorem Proof – Method 01

Let’s build the proof through an example.

Draw a right-angled triangle **ABC** where **AB** = 4 cm, **AC** = 3 cm and **BC** = 5 cm

By looking at the figure we can see that,

**Area of the square on BC = Area of the square on AB + Area of the square on AC**

**25 = 16 + 9 = 25**

Accordingly

Area of the square drawn on **PQ** = **a ^{2}**

Area of the square drawn on

**PR**=

**b**

^{2}Area of the square drawn on

**QR**=

**c**

^{2}Therefore,

**c ^{2} = a^{2} + b^{2}**

This is called the **Pythagorean theorem**.

### The Pythagorean Theorem Proof – Method 02

The figure to the proper indicates one among the various known proofs of this fundamental result. Indeed, the area of the “big” square is **(a + b) ^{2}** and can be decomposed into the area of the smaller square plus the areas of the four congruent triangles.

That is,

**(a + b) ^{2} = c^{2} + 2ab**

which immediately reduces to **a ^{2} + b^{2} = c^{2}**

## Pythagorean Triples

Pythagorean Triples are three positive integer values which satisfy the Pythagoras theorem c^{2} = a^{2} + b^{2}

Below table includes few Pythagorean triples examples.

a^{2} | b^{2} | c^{2} |

3 | 4 | 5 |

6 | 8 | 10 |

7 | 24 | 25 |

5 | 12 | 13 |

20 | 21 | 29 |

8 | 15 | 17 |

20 | 99 | 101 |

48 | 55 | 73 |

17 | 144 | 145 |

9 | 40 | 41 |

11 | 60 | 61 |

12 | 35 | 37 |

13 | 84 | 85 |

16 | 63 | 65 |

28 | 45 | 53 |

33 | 56 | 65 |

36 | 77 | 85 |

39 | 80 | 89 |

48 | 55 | 73 |

65 | 72 | 97 |

## Pythagorean Theorem Calculator

** Exercises:**

**01.**

Prove Euclid’s Theorem for Proportional Segments, i.e., given the right triangle ΔABC as indicated, then **h ^{2} = pq, a^{2} = pc, b^{2} = qc**

**02.** Prove that the sum of the interior angles of a quadrilateral ABCD is 360^{◦}.

**03.**

In the diagram to the right, ΔABC is a right triangle, segments [AB] and [AF] are perpendicular and equal in length, and [EF] is perpendicular to [CE]. Set a = BC, b = AB, c = AB, and deduce President Garfield’s proof* of the Pythagorean theorem by computing the area of the trapezoid BCEF.

*James Abram Garfield (1831–1881) published this proof in 1876 in the Journal of Education (Volume 3 Issue 161). While a member of the House of Representatives. He was assassinated in 1881 by Charles Julius Guiteau. As an aside, notice that Garfield’s diagram also provides a simple proof of the fact that perpendicular lines in the planes have slopes which are negative reciprocals.

Related lessons: Trigonometric Identities

## FAQ about Pythagorean Theorem

### What is the formula of Pythagoras?

The theorem, also referred to as the Pythagorean theorem, states that the square of the length of the hypotenuse is adequate** **to** **the sum of squares of the lengths of the opposite** **two sides of the right triangle.

**Hypotenuse ^{2} = Perpendicular^{2} + Base^{2}**

### What is meant by Pythagoras Theorem?

The Pythagoras theorem also referred to as Pythagorean theorem is employed to seek out the sides of a right angled triangle. This theorem is usually utilized in Trigonometry, where we use trigonometric ratios like sine, cos, tan to seek out the length of the edges of the right triangle

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OctavianI find very useful also the Reciprocal Pythagorean theorem. (The one that says if, in a triangle, the sum of the squares of two sides equals the square of the other side => the triangle is right-angled,