The concept of Pythagoras theorem has been around for over 2000 years and is credited to an ancient Greek mathematician named Pythagoras and his group, the Brotherhood of Pythagoras. They made significant contributions to the field of mathematics, particularly in the development of an algebraic method used in geometry.

Pythagoras theorem is a fundamental mathematical principle that explains the relationship between the sides of a right triangle. It states that in a right triangle, the square of the longest side (known as the hypotenuse) is equal to the sum of the squares of the other two sides. In simpler terms, the area of the largest square is equal to the combined area of the two smaller squares.

Visual representation of Pythagoras theorem using squares and a right triangle:

- Insert image of squares and right triangle for visual aid

The formula for Pythagoras theorem can be written as follows:

**hypothenuse ^{2} = opposite^{2} + adjacent^{2}**

Using this formula, we can determine the missing side of a right triangle when the other two sides are known. For example, if we are given a right triangle with the opposite and adjacent sides measuring 3 and 4 units respectively, and we need to find the hypotenuse, we can use the following steps:

**Hypothenuse ^{2} = 3^{2} + 4^{2}**

**Hypothenuse ^{2} = 9 + 16**

**Hypothenuse ^{2} = 25**

**Hypothenuse = √25**

**Hypothenuse = 5**

Therefore, the missing side (hypotenuse) of the right triangle measures 5 units.

A set of three numbers that satisfy the conditions of Pythagoras theorem is known as a Pythagorean triple. In other words, the square of the largest number in the set is equal to the sum of the squares of the other two numbers in the set. For example, 3, 4, and 5 is a Pythagorean triple because 5^{2} = 3^{2} + 4^{2}. However, 7, 12, and 5 is not a Pythagorean triple because 12^{2} ≠ 7^{2} + 5^{2}.

We can verify if a set of numbers is a Pythagorean triple using the same steps as solving for a missing side using Pythagoras theorem.

**Is 7, 12, and 5 a Pythagorean triple?**

**12 ^{2} = 7^{2} + 5^{2}**

**144 = 49 + 25**

**144 = 74**

As the equation does not hold, 7, 12, and 5 are not a Pythagorean triple.

**Is 8, 15, and 17 a Pythagorean triple?**

**8 ^{2} = 15^{2} + 17^{2}**

**64 = 225 + 289**

**64 = 514**

As the equation holds, 8, 15, and 17 are a Pythagorean triple.

- Pythagoras theorem helps explain the relationship between the sides of a right triangle and can assist in solving for missing sides.
- The formula for Pythagoras theorem is hypothenuse
^{2}= opposite^{2}+ adjacent^{2}. - The longest side of a right triangle is known as the hypotenuse, while the other two sides are called the opposite and adjacent.
- A Pythagorean triple is a set of three numbers that satisfy Pythagoras theorem's conditions.

**What is Pythagoras theorem?**Pythagoras theorem is a fundamental principle that explains the relationship between the sides of a right-angled triangle.**What is the hypotenuse in Pythagorean theorem?**The hypotenuse in Pythagorean theorem is the longest side of the right triangle.**How can Pythagoras theorem be applied?**Pythagoras theorem is used to find the missing side of a right triangle when the other two sides are known.**What is the formula for Pythagoras theorem?**The formula for Pythagoras theorem is hypothenuse^{2}= opposite^{2}+ adjacent^{2}.

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