Pythagorean Theorem (Definition, Types and Examples) - BYJUS

# Pythagorean Theorem

The Pythagorean theorem is a fundamental theorem in geometry that was proposed by the Greek philosopher Pythagoras. The theorem describes a relationship between the sides of a right-angled triangle. This theorem has found applications in various fields of study where geometry is involved....Read MoreRead Less ## Pythagorean Theorem

Pythagoras was a Greek mathematician and philosopher. He lived in Samos during the 6th century B.C. He proved a very important rule of mathematics related to right-angled triangles. A theorem is a rule in math that has logical proof. Since Pythagoras successfully proved the theorem related to right-angled triangles, it is called the Pythagorean theorem.

The Pythagorean theorem is among the most important topics of mathematics, and it describes the relationship between the sides of a right-angled triangle.

The core aspect of this theorem is to find the measure of an unknown length or an unknown angle of a right-angled triangle. We can derive formulas on the base, height and hypotenuse of a triangle with the help of this theorem. Let’s understand this theorem in detail.

## Pythagorean Theorem Statement

The Pythagorean theorem states that “In any right-angled triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse”. The legs of this triangle are the shorter sides and the longest side, opposite the 90-degree angle, is called the hypotenuse.

Sides a and b form the legs and side c forms the hypotenuse of the right triangle. ## Pythagorean Theorem Formula

For the triangle shown below, as per the definition of the theorem, the Pythagorean theorem formula can be given as:

$$a^2 + b^2 = c^2$$

Let us understand this with the help of a small activity.

Draw a right-angled triangle with one side along the horizontal and the second side along the vertical on a grid paper.

Mark the measure of the two shorter sides as a and b and the measure of the longest side as c. Draw 3 squares each sharing 1 side of the triangle as their sides.

Since we have 3 squares with sides a unit, b unit and c unit, respectively. So their corresponding area will be $$a^2$$ unit sq., $$b^2$$ unit sq. and $$c^2$$ unit sq. (as labelled in the diagram).

Add the areas of the two small squares. It is observed that this area is the same as the area of the largest square.

That is $$a^2 + b^2 = c^2$$

This is true for all right angled triangles.

So, $$a^2 + b^2 = c^2$$ is the basic principle of the Pythagoras theorem, where a and b are the measure of sides or legs, and c is the measure of the hypotenuse of any right angled triangle.

### Applications

We can use the Pythagorean theorem to:

1. To determine whether a triangle is right-angled or not.
2. To find the length of unknown sides in a right-angled triangle and three-dimensional figures.
3. To find the distance between two points on a coordinate plane.

## Solved Examples :

Example 1: Find the length of the Hypotenuse of the triangle. Solution:

$$a^2 + b^2 = c^2$$                    Write the Pythagorean theorem

$$(3)^2 + (4)^2 = c^2$$              Substitute 3 for a and 4 for b

$$9+16=c^2$$                      Evaluate the power

$$25=c^2$$                            Add.

$$5=c$$                               Take the positive square root of each side

The length of the hypotenuse is 13 inches.

Example 2: Find the length of the hypotenuse of the triangle. Solution:

$$a^2+b^2=c^2$$                     Write the Pythagorean Theorem

$$(8)^2+(15)^2=c^2$$             Substitute 3 for a and 4 for b

$$64+225=c^2$$                   Evaluate the power

$$289=c^2$$                            Add.

$$17=c$$                               Take the positive square root of each side.

The length of the hypotenuse is 17 feet.

Example 3: Find the missing length of the triangle. Solution:

$$a^2+b^2=c^2$$                    Write the Pythagorean Theorem

$$(2.4)^2+(x)^2=(7.4)^2$$    Substitute 2.4 for a, x for b and 7.4 for c

$$5.76+x^2=54.76$$           Evaluate the power

$$x^2=49$$                            Subtract 5.76 from both sides.

$$x=7$$                               Take the positive square root of each side

The missing length is 7 mi.

Example 4: Find the slant height of the triangular pyramid. Solution:

$$a^2+b^2=c^2$$                     Write the Pythagorean Theorem

$$(8.8)^2+(6.6)^2=(x)^2$$      Substitute 8.8 for a,  6.6 for b and x for c

$$77.44+43.56=x^2$$           Evaluate the power

$$121=x^2$$                            Add.

$$x=11$$                               Take the positive square root of each side

The length of the slant height is 11 ft.

Example 5:  Two ships are surveying around a lighthouse. Ship 1 is 50 feet north and 20 feet east of the light house. Ship 2 is 80 feet north and 60 feet east of the light house. How far is ship 1 from ship 2? Solution:

To solve the problem we first draw the situation in the coordinate plane. Let the origin represent the lighthouse location. From the descriptions, ship 1 is at (20, 50) and ship 2 is at (60, 80).

Then draw a right triangle with a hypotenuse that represents the distance between ship 1 and ship 2. The lengths of the legs are 30 feet and 40 feet.

Now use the Pythagorean theorem to find the length of the hypotenuse.

$$a^2+b^2=c^2$$                    Write the Pythagorean Theorem

$$(30)^2+(40)^2=c^2$$          Substitute 30 for a and 40 for b

$$900+1600=c^2$$              Evaluate the power

$$2500=c^2$$                         Add.

$$50=c$$                              Take the positive square root of each side.

So, ship 1 is 50 feet away from ship 2.