What is Pythagoras theorem - Class 7 PDF Download

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).”

a2+b2=c2

In the given triangle, side “a” is called as “Perpendicular”, side “b” is called as “Base” and side “c” is called as “Hypotenuse.”

The side opposite to the right angle (90∘) is the longest side known as Hypotenuse, as we know that the side opposite to the greatest angle is longest.

Another way of representation of the Pythagoras Theorem-

Another way of Pythagoras Theorem proof is described here. Consider a right-angled triangle having perpendicular as a, base as b, and hypotenuse as c. 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

Example- Prove the Pythagoras Theorem for a right angle triangle having sides to be 3cm, 4cm and 5 cm.Solution –

From Pythagoras Theorem we have,

(Perpendicular)2+(base)2=(Hypotenuse)2

Perpendicular = 3 cm

Base = 4 cm

Hypotenuse = 5 cm

(3)2+(4)2=(5)2

⇒9+16=25

⇒25=25

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

Therefore Pythagoras theorem is proved.

Example- The sides of a triangle are 5,12 & 13 units. Check if it has a right angle or not.Solution-

To prove- Pythagoras theorem in order to find whether it has a right angle or not

From Pythagoras Theorem, we have-

(Perpendicular)2+(base)2=(Hypotenuse)2

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units

(12)2+(5)2=(13)2

⇒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.

Example- 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

From Pythagoras Theorem, we have

(Perpendicular)2+(base)2=(Hypotenuse)2

152+b2=172

⇒225+b2=289

⇒b2=289–225

⇒b2=64

⇒b=64−−√

Therefore b=8

Example- 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)2=(ac)2

⇒(4)2+(4)2=(ac)2

⇒16+16=(ac)2

⇒32=(ac)2

⇒(ac)2=32

or acl=42–√