# Pythagoras Theorem

Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. This theorem is better explained with proof and formula. Students will come across many questions in Geometry where this theorem is applicable. The sides of a right triangle (say x, y and z) which has positive integer values, when squared are put into an equation, also called a Pythagorean triple.

## Pythagoras Theorem Statement

In a right-angled triangle, “the sum of squares of the lengths of the two sides is equal to the square of the length of the hypotenuse (or the longest side).”

In the given triangle, side “a” is the “Perpendicular”, side “b” is the “Base” and side “c” is the “Hypotenuse.” Then according to the definition, the Pythagoras Theorem formula is given as

Hypotenuse2 = Perpendicular2 + Base2

c2 = a2 + b

The side opposite to the right angle (90°)  is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

Consider three squares of sides a,b,c mounted on the three sides of a triangle having the same sides as shown.

By Pythagoras Theorem –

Area of square A + Area of square B = Area of square C

Note- Pythagorean theorem is only applicable to Right-Angled triangle.

## Pythagoras Theorem Proof

Given: A right-angles triangle ABC.

To Prove- AC2 = AB2 + BC2

Proof: First, we have to drop a perpendicular BD onto the side AC

Therefore, $\frac{AD}{AB}=\frac{AB}{AC}$ (Condition for similarity)

Or, AB= AD × AC ……………………………..……..(1)

Also, △BDC ~△ABC

Therefore, $\frac{CD}{BC}=\frac{BC}{AC}$ (Condition for similarity)

Or, BC2= CD × AC ……………………………………..(2)

Adding the equations (1) and (2) we get,

AB+ BC= AD × AC + CD × AC

AB+ BC= AC (AD + CD)

Since, AD + CD = AC

Therefore, AC2 = AB2 + BC2

Hence, the Pythagorean thoerem is proved.

## 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.

### Pythagorean Theorem Problems

Problem 1: The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.

Solution: From Pythagoras Theorem, we have;

Perpendicular2 + Base2 = Hypotenuse2

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

122 + 52 = 132

⇒ 144 + 25 = 169

⇒ 169 = 169

L.H.S. = R.H.S.

Therefore, the angles opposite to the 13 unit side will be at a right angle.

Problem 2: The two sides of a right-angled as shown in the figure. Find the third side.

Solution: Given;

Perpendicular = 15cm

Base = b cm

Hypotenuse = 17 cm

As per the Pythagorean Theorem, we have;

Perpendicular2 + Base2 = Hypotenuse2

⇒152 + b2 = 172

⇒225 + b2 = 289

⇒b2 = 289 – 225

⇒b2 = 64

⇒b = √64

Therefore, b = 8

Problem 3: Given the side of a square to be 4 cm. Find the length of the diagonal.

Solution- Given;

Sides of a square = 4 cm

To Find- The length of diagonal ac.

Consider triangle abc (or can also be acd)

(ab)2 +(bc)= (ac)2

(4)2 +(4)2= (ac)2

16 + 16 = (ac)2

32 = (ac)2

(ac)2 = 32

or ac = 4√2.

Thus, the length of the diagonal is 4√2.

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 Related Links Right Angled Triangle Constructions Rhs Right Angle Triangle Theorem Right Triangle Congruence Theorem

#### Practise This Question

In the figure ABC is a right angled triangle with right angle at B. BD is perpendicular to AC. Then which of the following options will hold true?