In this chapter we will learn the concept of Pythagoras theorem with solved examples.

## What is Pythagoras Theorem ?

It gives a formula to find side length of right angles triangle.

According to the formula, **the square of hypotenuse length is equal to sum of square of other sides of right triangle**.

Given above is right angled triangle with ∠ABC = 90 degree and AC as hypotenuse.**According to Pythagoras theorem;**

\mathtt{hypotenuse^{2} =\ side^{2} +\ side^{2}}

\mathtt{AC^{2} =\ AB^{2} +\ BC^{2}}

The above concept and formula is very important and is used in various math topics. Make sure you remember this formula for your examination.

I hope you understood the concept, let us solve some problems for better understanding.

## Pythagoras theorem formula – Solved Examples

(01) Given below is the right angle triangle ABC. Find the length of side AC.

**Solution**

Since ABC is a right angled triangle, we can apply Pythagoras theorem formula.

\mathtt{AC^{2} =\ AB^{2} +\ BC^{2}}

Putting the values, we get;

\mathtt{AC^{2} =\ 4^{2} +\ 3^{2}}\\\ \\ \mathtt{AC^{2} =\ 16\ +\ 9}\\\ \\ \mathtt{AC^{2} =\ 25}\\\ \\ \mathtt{AC\ =\ \sqrt{25}}\\\ \\ \mathtt{AC\ =\ 5}

Hence,** length of side AC = 5 cm.**

(02) Given below is rectangle ABCD with side length = 5 cm and diagonal length = 13 cm. Find the length of breadth AB.

**Solution**

Since the rectangle ABCD is a rectangle, the measure of all angles is 90 degree.

So, ∠ABC = 90 degree.

Applying Pythagoras theorem formula in triangle ABC.

\mathtt{AC^{2} =\ AB^{2} +\ BC^{2}}

Putting the values in formula.

\mathtt{13^{2} =\ AB^{2} +\ 5^{2}}\\\ \\ \mathtt{169\ =\ AB^{2} \ +\ 25}\\\ \\ \mathtt{AB^{2} =\ 169\ -\ 25}\\\ \\ \mathtt{AB^{2} =\ 144}\\\ \\ \mathtt{AB\ =\ \sqrt{144}}\\\ \\ \mathtt{AB\ =\ 12\ cm}

Hence, **side of rectangle AB measure 12 cm.**

(03) In the below triangle ABC, BC = 12 cm, AB = 10 cm and AD is the perpendicular bisector of side BC. Find the length of side BC.

**Solution**

Since AD is perpendicular bisector it will divide side BC into two equal parts.

So, BD = DC = 12 / 2 = 6 cm

Also, ∠ADB = 90 degree.

Applying Pythagoras formula in triangle ADB.

\mathtt{AB^{2} =\ BD^{2} +\ AD^{2}}\\\ \\ \mathtt{10^{2} =\ 6^{2} +\ AD^{2}}\\\ \\ \mathtt{100\ =\ 36\ +\ AD^{2}}\\\ \\ \mathtt{AD^{2} =\ 100\ -\ 36}\\\ \\ \mathtt{AD^{2} =\ 64}\\\ \ \mathtt{AD\ =\ \sqrt{64}}\\\ \\mathtt{AD\ =\ 8\ cm}

Hence, **length of side AD = 8 cm**.

(04) Observe the below image and find length of side CD.

Solution

From vertex D, draw perpendicular line to side BC as shown below.

Triangle DOC is a right triangle.

Using Pythagoras theorem.

\mathtt{DC^{2} =\ OD^{2} +\ OC^{2}}\\\ \\ \mathtt{DC^{2} =\ 12^{2} +\ 9^{2}}\\\ \\ \mathtt{DC^{2} \ =\ 144\ +\ 81}\\\ \\ \mathtt{DC^{2} =\ 225}\\\ \\ \mathtt{DC\ =\ \sqrt{225}}\\\ \\ \mathtt{DC\ =\ 15\ cm}

Hence,** length of side DC = 15 cm.**

(05) A boy traveled 6 km towards north then right and move 8 km towards east. What is the shortest distance between initial and final position.**Solution**

The shortest distance between any two point is a straight line.

Observe the below image.

Here point A is the starting position and B is the final location.

The line AB is the shortest distance between initial and final location.

Note that AOB form a right angle triangle.

Using Pythagoras theorem.

\mathtt{AB^{2} =\ OA^{2} +\ OB^{2}}\\\ \\ \mathtt{AB^{2} =\ 6^{2} +\ 8^{2}}\\\ \\ \mathtt{AB^{2} \ =\ 36\ +\ 64}\\\ \\ \mathtt{AB^{2} =\ 100}\\\ \\ \mathtt{AB\ =\ \sqrt{100}}\\\ \\ \mathtt{AB\ =\ 10\ cm}

Hence, **length of side AB is 10 cm.**