Pythagoras Theorem - Formula, Proof, Examples

Another famous name in mathematics is the Pythagoras theorem. Named after the ancient Greek mathematician Pythagoras, it has always been a crucial idea in geometry and is used extensively in problem-solving, construction, architecture, and today, in hi-tech applications. This blog will discuss the Pythagoras theorem formula, proof, examples, and give a simplified version of the Pythagorean theorem for students.

What Is the Pythagoras Theorem?

Definition:

Pythagoras theorem, in a right-angled triangle, states that the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides.

This property can be expressed in its very simple formula.

Pythagoras Theorem Formula

If a right triangle has:

Base = a

Perpendicular (height) = b

Hypotenuse = c (longest side)

Then the formula reads as:

a² + b² = c²

Pythagoras Theorem

Imagine that you make squares on each of the three sides of a right triangle: the two smaller squares (on a and b) together will have an area equal to the larger square on c.

If one side were 3 units and the other 4, then:

a² + b² = 3² + 4² = 9 + 16 = 25

The hypotenuse c = √25 = 5

Therefore, a 3-4-5 triangle is an ideal right triangle.

Proving the Pythagorean Theorem

There are countless proofs of this theorem—algebraic proofs, geometric proofs, and visual proofs. Here is a very simple proof from the geometric side:

Draw a square with a side length of (a + b).

Next, situate within this square four copies of the right triangle with legs a and b and hypotenuse c.

The empty area in the middle will be a small square with a side c.

Now find the area of the big square in two different ways:

From the formula: Area = (a + b)²;

From the parts: Area = 4 × (½ab) + c² = 2ab + c².

Expanding the first expression yields: (a + b)² = a² + 2ab + b².

Set them equal: a² + 2ab + b² = 2ab + c².

Cancel the 2ab terms: a² + b² = c².

Steps to prove Pythagoras theorem explained in a clear cut manner.

Pythagoras Theorem Examples

Let us apply the formula to a few problems.

Example 1:

A ladder leans against a wall. The foot of the ladder is 9 m away from the wall, while the ladder reaches up to 12 m high. Find the length of the ladder.

a = 9, b = 12

c² = a² + b² = 9² + 12² = 81 + 144 = 225

c = √225 = 15 m

So, the ladder is 15 meters long.

Example 2:

Hypotenuse of a right triangle is 13 cm with one side as 5 cm. Find the other side.

c = 13, a = 5, b = ?

b² = c² - a² = 13² - 5² = 169 - 25 = 144

b = √144 = 12 cm

So, the missing side is 12 cm.

Example 3:

Verify if the triangle with side measurements 7 cm, 24 cm, and 25 cm is right-angled in nature.

a = 7, b = 24, c = 25

a² + b² = 49 + 576 = 625

c² = 25² = 625

Hence, this triangle is a right triangle.

Some Applications of Pythagoras Theorem

This theorem is not only theory in a book but comes with several applications:

Construction and Architecture → To make sure the walls are perfectly perpendicular.

Navigation → To find shortest paths (air, land, or sea).

Engineering → In the design of structures and machinery.

Computer Graphics → Distance calculations in 2D and 3D environments.

In everyday activities → Measuring diagonal distances, height of poles, and ladders leaning against walls.

Fun Fact about Pythagorean Triples

Numbers that satisfy the given theorem are called Pythagorean Triples. Probably the most famous of these are:

(3,4,5)

(5,12,13)

(7,24,25)

(8,15,17)

These triples are massively used for quick solving of geometry problems.

Conclusion

The Pythagoras theorem formula proof examples showcase that for any right-angled triangle, a² + b² = c².

Pythagorean theorem explained simply means the square of the hypotenuse is equal to the sum of the squares on the other two sides.

With everyday applications and plenty of examples, it remains one of the most useful theorems in mathematics.

Learning Pythagoras Theorem will help students in exams and further develop their ability to reason logically and visualize spatially.

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Article Author

Zarghoona Abid Khan

Zarghoona Abid Khan is an experienced and passionate English instructor at Bambinos.live. She is dedicated to empowering students through effective teaching methodologies. Zarghoona is a graduate in English literature and a certified TEYL instructor. She possesses the knowledge and expertise to foster an engaging learning environment. Zarghoona efficiently teaches reading, writing, and English as a second language. She fosters a stimulating and inclusive learning environment and implements innovative approaches to enhance student engagement. Additionally, she utilizes various teaching resources and technologies effectively.

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