Geometry: Pythagorean Theorem

In this lesson, we will learn

• the Pythagorean Theorem
• the Converse of the Pythagorean Theorem
• how to use the Pythagorean Theorem
• proofs of the Pythagorean Theorem

The Pythagorean Theorem

A right triangle consists of two sides called the legs and one side called the hypotenuse. The hypotenuse is the longest side and is opposite the right angle.

The Pythagorean Theorem or Pythagoras' Theorem is a formula relating the lengths of the three sides of a right triangle.

If we take the length of the hypotenuse to be c and the length of the legs to be a and b then this theorem tells us that:

c² = a² + b²

Pythagorean theorem states that

In any right triangle, the sum of the squared lengths of the two legs is equal to the squared length of the hypotenuse.

Note: Pythagorean theorem only works for right triangles

The following video gives the definition of Right Triangle and Pythagorean Theorem

Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem is also true.

For any triangle with sides a, b, c, if a² + b² = c², then the angle between a and b measures 90° and the triangle is a right triangle.

We can use the converse of the Pythagorean Theorem to check whether a given triangle is an acute triangle, a right triangle or an obtuse triangle.

For a triangle with sides a, b and c and c is the longest side then:
If c² < a² + b² then it is an acute-angled triangle, i.e. the angle facing side c is an acute angle.

If c² = a² + b² then it is a right-angled triangle, i.e. the angle facing side c is a right angle.

If c²> a² + b² then it is an obtuse-angled triangle, i.e. the angle facing side c is an obtuse angle.

Pythagorean Theorem Worksheets

How to use the Pythagorean Theorem

The Pythagorean Theorem can be used when we know the length of two sides of a right triangle and we need to get the length of the third side.

Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 3 inches and 4 inches.

Solution:
Step 1: Write down the formula	c2 = a2 + b2
Step 2: Plug in the values	c2 = 32 + 42
	c2 = 9 + 16
	c2 = 25
	c =
	c = 5

Answer: The length of the hypotenuse is 5 inches.

Example 2: Find the length of one side of a right triangle if the length of the hypotenuse is 10 inches and the length of the other side is 9 inches.

Solution:
Step 1: Write down the formula	c2 = a2 + b2
Step 2: Plug in the values	102 = 92 + b2
	100 = 81 + b2
Step 3: Subtract 81 from both sides	19 = b2
	b =
	b » 4.36

Answer: The length of the side is 4.36 inches.

Have a look at the following video for more examples on how to use the Pythagorean theorem.

Proofs of the Pythagorean Theorem

There are many ways to proof the Pythagorean Theorem. We will look at three of them here.

Proof of the Pythagorean Theorem using Similar Triangles

This proof is based on the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Proof of the Pythagorean Theorem by Algebra

In this proof, we use four copies of the right triangle, rearrange them and use algebra to proof the theorem

Proof of the Pythagorean Theorem by Rearrangement

The following video shows how a square with area c² can be cut up and rearranged such that it can fit into two other smaller squares with areas a² and b².

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