# Pythagoras Theorem in Geometry

## What is Pythagoras Theorem?

Pythagoras Theorem is a fundamental theorem in Geometry, named after the famous Greek mathematician Pythagoras. It states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is also known as the Pythagorean Theorem.## How to Prove the Pythagoras Theorem?

The Pythagoras Theorem can be proved using two different methods. The first method involves constructing a square with the length of the hypotenuse as its side. The second method is to use the Pythagorean Triples, which are a set of special right triangles whose sides are in the ratio of whole numbers.## What are the Applications of Pythagoras Theorem?

The Pythagoras Theorem has a wide range of applications in mathematics, engineering, and architecture. It is widely used to calculate the lengths of the sides of a right triangle, as well as the distances between two points. It is also used to calculate the area of a triangle and the volume of a pyramid or cone.## Solving Problems with Pythagoras Theorem

The Pythagoras Theorem can be used to solve a variety of problems related to right triangles. Let's look at some examples of how to use the theorem to solve problems.### Example 1: Find the length of the Hypotenuse

Let's say we have a right triangle with sides of length 3 and 4. We can use the Pythagoras Theorem to find the length of the hypotenuse:Hypotenuse^{2} = Side1^{2} + Side2^{2}

Hypotenuse^{2} = (3^{2}) + (4^{2})

Hypotenuse^{2} = 9 + 16

Hypotenuse^{2} = 25

Hypotenuse = v25 = 5

Therefore, the length of the hypotenuse is 5.### Example 2: Find the Area of a Triangle

Let's say we have a right triangle with sides of length 7, 24, and 25. We can use the Pythagoras Theorem to find the area of the triangle:Area = ï¿½(Base ï¿½ Height)

Area = ï¿½(7 ï¿½ 25)

Area = ï¿½(175)

Area = 87.5

Therefore, the area of the triangle is 87.5.### Example 3: Find the Length of the Third Side

Let's say we have a right triangle with sides of length 5 and 12. We can use the Pythagoras Theorem to find the length of the third side:Side3^{2} = Hypotenuse^{2} - Side1^{2} - Side2^{2}

Side3^{2} = (13^{2}) - (5^{2}) - (12^{2})

Side3^{2} = 169 - 25 - 144

Side3^{2} = 0

Side3 = v0 = 0

Therefore, the length of the third side is 0.## Practice Problems

- Find the length of the hypotenuse of a right triangle with sides of length 5 and 12.
- Find the area of a right triangle with sides of length 6, 8, and 10.
- Find the length of the third side of a right triangle with sides of length 3 and 4.
- Find the length of the hypotenuse of a right triangle with sides of length 8 and 15.
- Find the area of a right triangle with sides of length 5, 12, and 13.
- Find the length of the third side of a right triangle with sides of length 6 and 8.

Answer: 13

Answer: 24

Answer: 5

Answer: 17

Answer: 30

Answer: 10

## Summary

In this article, we discussed the Pythagoras Theorem, a fundamental theorem in Geometry named after the famous Greek mathematician Pythagoras. We looked at how to prove the theorem and its applications in mathematics, engineering, and architecture. We also looked at some examples of how to use the theorem to solve problems. Finally, we practiced some problems to help you better understand the theorem.## FAQ

### What is a Pythagorean theorem in geometry?

A Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

### What is Pythagoras theorem example?

For example, in a right triangle with sides of lengths 3, 4, and 5, the hypotenuse is 5 and the Pythagorean theorem states that 32 + 42 = 52, or 9 + 16 = 25.

### What is Pythagoras theorem as a formula?

Pythagoras theorem can be expressed as an equation: a2 + b2 = c2, where a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse.

### Who is Pythagoras short answer?

Pythagoras was a Greek philosopher and mathematician, born in the 6th century BC. He is best known for his contribution to geometry, particularly the Pythagorean theorem, which he is credited with discovering.