What is the Pythagoras Theorem?

The Pythagoras theorem states that if a triangle is a right-angled triangle, then the square of the hypotenuse is equal to the sum of the squares of the other two sides. Observe the following triangle ABC, in which we have BC² = AB² + AC². Here, AB is the base, AC is the altitude (height), and BC is the hypotenuse. It is to be noted that the hypotenuse is the longest side of a right-angled triangle.These theories can be understood better, with an online calculus tutor or online algebra tutoring who can guide you through these ideas step by step

Pythagoras Theorem Equation

The Pythagoras theorem equation is expressed as, c² = a² + b², where 'c' = hypotenuse of the right triangle and 'a' and 'b' are the other two legs. Hence, any triangle with one angle equal to 90 degrees produces a Pythagoras triangle and the Pythagoras equation can be applied in the triangle.

1. The Historical Background

Even though the theorem has Pythagoras’ name attached to it, the story doesn’t really begin with him. Pythagoras lived in Greece around 570–495 BCE, but there’s good evidence that the basic idea of his theorem was already known to other civilizations centuries earlier.

1.1 The Babylonians

Archaeologists discovered clay tablets from as far back as 1800 BCE covered in rows of numbers. One of the most famous of these is Plimpton 322, which listed what we now call Pythagorean triples like (3, 4, 5). Nearly 1,200 years before Pythagoras, people were already using this principle.

1.2 Ancient India

The Sulba Sutras (800–500 BCE), practical manuals for building fire altars, included instructions requiring right angles and diagonal measurements. These show the Pythagoras Theorem in action.

1.3 Ancient China

In the Zhou Bi Suan Jing (around 1000 BCE), the principle was called the Gougu rule. It was applied in land measurement, construction, and astronomy.

So, when we say “Pythagoras’ Theorem,” we’re really talking about a truth known across many cultures. What Pythagoras’ school contributed was the first rigorous proof, which is why his name is remembered.

Proofs of the Theorem

Over centuries, mathematicians have come up with many clever proofs. Here are a few famous ones:

(a) The Rearrangement Proof

• Draw a big square with side (a + b).
  
• Place four identical right-angled triangles inside it, leaving a smaller square of side c.
  
• Compare areas:
  
  • Whole square: (a + b)²
    
  • Parts: 4 triangles (½ab each) + c²
    
• Simplify → a² + b² = c².
  

(b) Euclid’s Proof

Euclid built squares on all three sides of a right-angled triangle. The area of the square on the hypotenuse equaled the sum of the other two — a visual, cut-and-paste style proof. These patterns can be understood better, with an online calculus tutor or online algebra tutoring who can guide you through these ideas step by step.

(c) President Garfield’s Proof

In 1876, U.S. President James Garfield devised a proof using the area of a trapezoid made from three right triangles. The calculation neatly produced a² + b² = c².

(d) Algebraic / Coordinate Proof

Put a right triangle on the coordinate plane: one corner at the origin, one along the x-axis, and one along the y-axis. Use the distance formula for the hypotenuse, and it directly gives a² + b² = c². Working with a 3rd grade tutoring online or 4th grade math tutor can help younger students understand these proofs intuitively

Pythagoras Theorem Formula

We frequently discuss this in our online math tutoring sessions at Ruvimo. The Pythagorean theorem formula states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the squares of the other two legs.

If AB and AC are the sides and BC is the hypotenuse of the triangle, then:
BC² = AB² + AC²

Another way to understand the Pythagorean theorem formula is using the following figure which shows that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle.

Pythagorean Theorem Formula Proof using Similar Triangles

Two triangles are said to be similar if their corresponding angles are of equal measure and their corresponding sides are in the same ratio.

Derivation:

• Consider a right-angled triangle ABC, right-angled at B. Draw a perpendicular BD meeting AC at D.
  
• △ABD ∼ △ACB and △BCD ∼ △ACB.
  
• Using similarity, we get AB² + BC² = AC².
  
• Hence proved.
  

Pythagoras Theorem Triangles

Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The three sides of such a triangle are collectively called Pythagoras triples.

Examples: (3, 4, 5), (5, 12, 13).

These patterns can be understood better, with an online calculus tutor or online algebra tutoring who can guide you through these ideas step by step.

Pythagoras Theorem Squares

As per the Pythagorean theorem, the area of the square which is built upon the hypotenuse of a right triangle is equal to the sum of the area of the squares built upon the other two sides.

Applications of Pythagoras Theorem

The applications of the Pythagoras theorem can be seen in our day-to-day life.

• Engineering and Construction – architects use it to calculate unknown dimensions.
  
• Face Recognition – security cameras use distance calculations.
  
• Woodwork and Interior Design – applied in layouts of furniture and houses.
  
• Navigation – sailors use it to find the shortest route across the sea.
  

Our US math tutors love showing students how math shows up in creative spaces. For many learners, it’s a total game-changer.

Applications of Pythagoras Theorem

• To know if the triangle is a right-angled triangle or not.
• In a right-angled triangle, we can calculate the length of any side if the other two sides are given.
• To find the diagonal of a square.

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side. If you're curious about how patterns like this work or want to dive deeper into the beauty of math, an online math tutor or a calculus tutor online can help you explore these fascinating concepts.

How to use Pythagoras Theorem?

To use Pythagoras theorem, remember the formula given below:

c2 = a2 + b2

Where a, b and c are the sides of the right triangle.

For example, if the sides of a triangles are a, b and c, such that a = 3 cm, b = 4 cm and c is the hypotenuse. Find the value of c.

We know, 

c2 = a2 + b2

c2 = 32+42

c2 = 9+16

c2 = 25

c = √25

c = 5 cm

Hence, the length of hypotenuse is 5 cm.

How to find whether a triangle is a right-angled triangle?

If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem.

Let us understand this statement with the help of an example.

Suppose a triangle with sides 10cm, 24cm, and 26cm are given. 

Clearly, 26 is the longest side.

It also satisfies the condition, 10 + 24 > 26

We know,

c2 = a2 + b2    ………(1)

So, let a = 10, b = 24 and c = 26

First we will solve R.H.S. of equation 1.

a2 + b2 = 102 + 242 = 100 + 576 = 676

Now, taking L.H.S, we get;

c2 = 262 = 676

We can see, 

LHS = RHS

Therefore, the given triangle is a right triangle, as it satisfies the Pythagoras theorem. If you're curious about how patterns like this work or want to dive deeper into the beauty of math, an online math tutor or a calculus tutor online can help you explore these fascinating concepts.

When is Pythagoras Theorem used?

Pythagoras theorem is used when any two sides of a right-angled triangle are given and the third side needs to be calculated. For example, if the perpendicular and base of a right-angled triangle are given as 12 units and 5 units respectively, and we need to find the third side (the hypotenuse) we can calculate it using the theorem which says hypotenuse2 = perpendicular2 + base2. After substituting the values in the equation we get hypotenuse2 = 122 + 52 = 144 + 25 = 169. So, hypotenuse = √169 = 13 units. That’s why we include these kinds of insights in our online math tutoring programs.

Practice questions

Example 1: The hypotenuse of a right-angled triangle is 16 units and one of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula.
Solution:
Given: Hypotenuse = 16 units
Let us consider the given side of a triangle as the perpendicular height = 8 units
On substituting the given dimensions to the Pythagoras theorem formula
Hypotenuse² = Base² + Height²
16² = B² + 8²
B² = 256 - 64
B = √192 = 13.856 units
Therefore, the measure of the third side of the triangle is 13.856 units. These patterns can be understood better, with an online calculus tutor or online algebra tutoring who can guide you through these ideas step by step.

Example 2: Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long. How far should she place the base of the ladder away from the building?
Solution:
We can visualize this scenario as a right triangle. We need to find the base of the right triangle formed. We know that, Hypotenuse² = Base² + Height². Thus, we can say that b² = 13² - 12² where 'b' is the distance of the base of the ladder from the feet of the wall of the building. So, b² = 13² - 12² can be solved as, b² = 169 - 144 = 25. This means, b = √25 = 5. Hence, we get 'b' = 5.
Therefore, the base of the ladder is 5 feet away from the building.

Example 3: Use the Pythagoras theorem to find the hypotenuse of the triangle in which the sides are 8 units and 6 units respectively.
Solution:
Using the Pythagoras theorem, Hypotenuse² = Base² + Height² = 8² + 6². This leads to Hypotenuse² = 64 + 36 = 100. Therefore, hypotenuse = √100 = 10 units.
Therefore, the length of the hypotenuse is 10 units.

How to find whether a triangle is a right-angled triangle?

If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem.

Let us understand this statement with the help of an example.

Suppose a triangle with sides 10cm, 24cm, and 26cm are given. 

Clearly, 26 is the longest side.

It also satisfies the condition, 10 + 24 > 26

We know,

c2 = a2 + b2    ………(1)

So, let a = 10, b = 24 and c = 26

First we will solve R.H.S. of equation 1.

a2 + b2 = 102 + 242 = 100 + 576 = 676

Now, taking L.H.S, we get;

c2 = 262 = 676

We can see, 

LHS = RHS

Therefore, the given triangle is a right triangle, as it satisfies the Pythagoras theorem.