CLASS 7TH | UNIT 4: GEOMETRY | PYTHAGORAS THEOREM | REVISION NOTES

Pythagoras Theorem Statement

Pythagoras theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular,

“b” is the base,

“c” is the hypotenuse.

According to the definition, the Pythagoras Theorem formula is given as:

Hypotenuse2 = Perpendicular2 + Base2c2 = a2 + b2

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.

Pythagoras Theorem Proof

Given: A right-angled triangle ABC, right-angled at B.

To Prove- AC² = AB² + BC²

Construction: Draw a perpendicular BD meeting AC at D.

Proof:

We know, △ADB ~ △ABC

Therefore, 

 (corresponding sides of similar triangles)

Or, AB² = AD × AC ……………………………..……..(1)

Also, △BDC ~△ABC

Therefore,

(corresponding sides of similar triangles)

Or, BC²= CD × AC ……………………………………..(2)

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

AB² + BC² = AD × AC + CD × AC

AB² + BC² = AC (AD + CD)

Since, AD + CD = AC

Therefore, AC² = AB² + BC²

Hence, the Pythagorean theorem 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.

Useful For

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

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

Let,

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units {since it is the longest side measure}

12² + 5² = 13²

⇒ 144 + 25 = 169

⇒ 169 = 169 

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

Therefore, the angle opposite to the 13 units side will be a right angle.

Problem 2: 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

ac = 4√2.

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