pythagoras' theorem

In this lesson, we will learn what is Pythagoras Theorem and how to use it.

First, let's look at a few definitions. 

Right triangle

A right triangle has one 90 degrees angle inside the triangle, which is called a right angle. Often, the right angle is shown with a box. 

Hypotenuse

In a right triangle, the hypotenuse is the longest side. It is the side directly across from the right angle. It is the only side of the triangle that is not a part of the right angle. 

Exponents

An exponent is a number that appears slightly above the right of another number like this: 2³. It is a quantity indicating the power to which a given number or expression is raised, as a symbol beside the number or expression (e.g. 2³ = 2 × 2 × 2). 

Now, we understand about the Pythagoras Theorem. 

Over 2000 years, an amazing discovery about triangles was made: 

When a triangle has a right angle (90^o) and squares are made on each of the three sides then the biggest square has the exact same area as the other two squares!

It is called "Pythagoras Theorem" and can be written in one short equation as: 

a² + b² = c²

where,

c is the longest side of the triangle.

a and b are the other two sides. 

As the longest side of the triangle is called the 'hypotenuse', the formal definition is: 

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 

The Pythagoras Theorem states the relationship between the sides of a right triangle, where c stands for the hypotenuse, and a and b are the sides forming the right angle. The formula is: 

a² + b² = c²

It is read ''a-squared plus b-squared equals c-squared.''

Let's see how it works. 

Example #1

Look at the following right-angled triangle with sides 3,4,5

⇒ 3² + 4² = 5²

⇒ 9 + 16 = 25

So, we see it works!

This is useful because if we know the lengths of two sides of a right-angled triangle, we can find the length of the third side. But remember it only works on the right-angled triangle!

Example #2

Let's solve for one more triangle below. Can you find out the value of c? 

⇒ 5² + 12² = c²

⇒ 25 + 144 = c²

⇒ 169 = c²

⇒ \(\sqrt{169}\) = c

⇒ 13 = c

So, the value of c is 13. 

Example #3

Let's look at another type of problem using the Pythagoras Theorem. 

Does the following triangle have a right angle? 

Apply the Pythagoras Theorem: 

⇒ a² + b² = c²

Solving for a² + b², we get

⇒ 10² + 24² = 100 + 576 = 676

Solving for  c²

⇒ c² = 26^{2 =} 676

They are equal, so YES this triangle has a right angle. 

What are the Pythagorean Triples? 

The Pythagorean Triples are the three integers used in the Pythagorean Theorem, which are a, b, and c. 

Why is the Pythagorean Theorem important? 

• It helps to find out if the triangle is a right angle. If the sum of two squared sides is equal to the squared value of the third side, which is the hypotenuse, then, the triangle is a right-angled triangle. 
• It also helps to find missing side lengths of a triangle. We can not only find the length of the third side of a right-angled triangle but also find the missing side lengths to squares and rectangles when the triangles are pushed together. Thus, the Pythagorean Theorem can help build rectangles and squares. 
• Builders use the Pythagorean Theorem to help keep the right angles and build houses, roofs, stairways, etc. 
• We may also use it in different aspects of our lives, for example, calculating the shortest distance between two points, when we are traveling. 

To wrap up, let's summarize what we've learned in this lesson.

The Pythagorean Theorem describes the relationships between the sides of a right triangle. The square of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares of the two sides. The formula is: a² + b² = c². We can determine whether or not the triangle is right-angle and also use the Pythagoras Theorem to find the missing side lengths of a right-angle triangle.