# Pythagoras Theorem – Formula, Proof, Examples

“The Pythagorean Theorem is like a special math rule that helps us with right triangles, which are triangles with one 90-degree angle, like a square corner. It helps us figure out the lengths of the sides of those triangles.

## What is the Pythagoras Theorem?

The Pythagorean Theorem states that in a right-angled 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 other two sides.

Imagine you have a right triangle with a shorter side ‘a’ and another shorter side ‘b.’ The Pythagorean Theorem says that if you add up the squares of these two sides and then take the square root, you’ll find the length of the longest side, which we call the ‘hypotenuse.’ We write it like this: a² + b² = c². So, it’s like a magic formula that helps us know how long that longer side is when we know the other two sides. This is really useful for building things, measuring distances, and even in some video games when you need to know how far your character can jump or run. It’s a cool math trick that makes triangles easier to understand!”

## Pythagorean Theorem Formula

The Pythagorean Theorem is a fundamental principle in geometry and is expressed with the following formula:

Perpendicular2 + Base2 = Hypotenuse2

a² + b² = c²

In pythagorean theorem formula:

• a and b represent the lengths of the two shorter sides of a right-angled triangle (the “legs”).
• c represents the length of the longest side, which is the side opposite the right angle and is called the “hypotenuse.”

The equation for pythagorean theorem is used to calculate side lengths or distances within right-angled triangles. It tells us that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse.

## Pythagorean Theorem Examples

Certainly! Here are six solved examples that demonstrate the application of the Pythagorean Theorem:

###### Example 1: In a right-angled triangle, one leg has a length of 3 units, and the other leg is 4 units long. What is the length of the hypotenuse?

Solution:
Using the Pythagorean Theorem: a² + b² = c²
Where a = 3, b = 4
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25
c = 5 units

###### Example 2: In a right-angled triangle, one leg measures 5 units, and the hypotenuse is 13 units long. What is the length of the other leg?

Solution:
Using the Pythagorean Theorem: a² + b² = c²
Where a = 5, c = 13
5² + b² = 13²
25 + b² = 169
b² = 169 – 25
b² = 144
b = √144
b = 12 units

###### Example 3: Determine if a triangle with sides of lengths 6, 8, and 10 units is a right triangle.

Solution:
Using the Pythagorean Theorem: a² + b² = c²
Where a = 6, b = 8, c = 10
6² + 8² = 10²
36 + 64 = 100
Since 100 = 100, this triangle is a right triangle.

###### Example 4: Two points in a Cartesian plane are located at (3, 4) and (6, 8). What is the distance between these two points?

Solution:
To find the distance, use the distance formula, which is derived from the Pythagorean Theorem:
Distance = √((x₂ – x₁)² + (y₂ – y₁)²
Distance = √((6 – 3)² + (8 – 4)²)
Distance = √(3² + 4²)
Distance = √(9 + 16)
Distance = √25
Distance = 5 units

###### Example 5: In a 3D space, you have a right-angled triangle with one side measuring 7 units, and the hypotenuse is 10 units long. What is the length of the other side in 3D space?

Solution:
Using the Pythagorean Theorem: a² + b² = c²
Where a = 7, c = 10
7² + b² = 10²
49 + b² = 100
b² = 100 – 49
b² = 51
b ≈ √51
b ≈ 7.14 units

###### Example 6: A rectangular room has dimensions of 6 feet by 8 feet. What is the length of the diagonal across the room?

Solution:
The diagonal of a rectangle can be found using the Pythagorean Theorem. Let a and b represent the lengths of the rectangle’s sides.
a = 6 feet, b = 8 feet
Using the Pythagorean Theorem: diagonal² = a² + b²
diagonal² = 6² + 8²
diagonal² = 36 + 64
diagonal² = 100
diagonal = √100
diagonal = 10 feet

## FAQs on Pythagoras Theorem

1. Who is Pythagoras, and why is this theorem named after him?

• Pythagoras was an ancient Greek mathematician and philosopher. This theorem is named after him because he and his followers, the Pythagoreans, are credited with its discovery and understanding.

2. What does the Pythagorean Theorem state?

• The theorem states that in a right-angled 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 other two sides.

3. How is the Pythagorean Theorem used in real life?

• The theorem is used in various real-life applications, including construction, architecture, navigation, and in computer graphics to calculate distances and angles.

4. Can the Pythagorean Theorem be applied to non-right triangles?

• No, the theorem is specific to right-angled triangles. It doesn’t apply to other types of triangles.
##### 5. How do you prove the Pythagorean Theorem?
• There are many different proofs of the theorem, including geometric, algebraic, and trigonometric methods. A common geometric proof involves constructing squares on the sides of a right triangle.

6. What is the formula for the Pythagorean Theorem?

• The formula is a² + b² = c², where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse.

7. How can the Pythagorean Theorem be used to find missing side lengths in a triangle?

• You can use the theorem to find a missing side length by rearranging the formula. For example, to find side a, you can use a = √(c² – b²).
###### 8. Are there variations of the Pythagorean Theorem for higher dimensions?
• Yes, there are generalizations of the theorem for higher dimensions, such as the Pythagorean Theorem in three-dimensional space.

9. What are some common mistakes when applying the Pythagorean Theorem?

• Common mistakes include using the theorem in non-right triangles, using the wrong formula, or not ensuring that the triangle is indeed right-angled.

10. Can the Pythagorean Theorem be applied to non-integer side lengths?

• Yes, the theorem can be applied to triangles with non-integer side lengths, including fractions and decimals. It’s a versatile mathematical tool.