[Maths Class Notes] on Pythagorean Theorem Pdf for Exam

Pythagoras of Samos was a Greek Philosopher and was considered to be the Son of a gem-engraver of an island of Samos. In the First Century, Pythagoras came up with a theorem that says, “the square of the hypotenuse of a right-angled triangle is equal to the sum of the square of the other two sides.”  This theorem is known as Pythagoras theorem. After discovering this theorem Pythagoras sacrificed an ox to God. 

 

Later, when this story started making a name for itself, one of the Pythagorean opponents named Cicero mocked the story saying that Pythagoras made the sacrifice of blood illegal then how can he sacrifice an ox. Porphyry, one of his followers, tried to justify the story claiming that the ox was made up of dough. 

 

It is also believed that the theorem discovered by Pythagoras was actually used by Babylonians 1000 of years before Pythagoras. The only difference was they used this theorem for an isosceles right-angled triangle where two sides of the right-angled triangle (i.e, base and height) were the same. Indian mathematicians also used this theorem from Sulbasutras which was written long before Pythagoras discovered the theorem. Apart from Indian, Chinese and Egyptian also used this theorem for various construction purposes.

 

Right-Angled Triangle and Pythagorean Theorem

What is a Right-Angled Triangle?

A right-angled triangle is a polygon of three sides having one angle as 90 degrees(right angle). In a right-angled triangle, the side opposite to the right angle is always bigger than the other two sides. This bigger side is called Hypotenuse, the side on which triangle rests is called base or adjacent and the third side is called height or perpendicular.

 

Right-Angled Triangle as a Combination of Three Squares

Suppose, you are given three squares such that two small squares are kept at 90degrees to each other and the side of the third square covers the open end in such a way that it makes a right-angled triangle. 

 

If the sides of two squares are given then you can find the side of the third square without actually measuring it. Suppose the sides of two smaller squares are 3cm and 4cm then you can find the side of the third square by using Pythagoras Theorem Model. Let’s see how. 

• Step 1: Let the square with sides 3 cm and 4 cm be square A and B respectively. Let the bigger square be C. 

• Step 2: Finding the area of squares A and B. 

Area of square A = side x side = 3 cm x 3 cm = 9 cm²

Area of square B = side x side = 4 cm x 4 cm = 16 cm²

Sum of the area of two smaller squares = Area of the bigger square.

Therefore, 

Area of Square A + Area of Square B = Area of Square C

9 cm² + 16 cm² = 25 cm²

Thus, the area of Square C = 25 cm²                                                                            

Area = (side)²

[Side = sqrt{Area} ]

[Side = sqrt{25 cm ^2} ]

Side = 5 cm.

 

Therefore, the side of square C is 5cm.  

 

Pythagorean Theorem Definition

Pythagoras theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the square of its base and height. 

 

(Hypotenuse)² = (Base)² + (Perpendicular)²

 

If the length of the base, perpendicular, and hypotenuse of a right-angle triangle is a, b and c respectively. Then, we can say: 

 

(a)² + (b)² = (c)²

 

This equation is also called a Pythagorean triple. Pythagoras theorem questions involve the application of the Pythagorean triple.

 

Geometrical Proof of Pythagorean Theorem 

Let us take a right-angled triangle ABC with angle B as 90 degrees. A line BD is drawn as perpendicular to the hypotenuse (i.e, AC). On comparing the triangle ADB and ABC we can say that, 

 

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Angle A of triangle ADB = Angle A of the triangle ABC

 

Angle D of triangle ADB = Angle B of the triangle ABC = 90 degrees

 

We know that the sum of three angles of a triangle is always 180 degrees thus if any two corresponding angles of two triangles are the same then the third corresponding angle of both the triangles is also the same. 

 

So, Angle B of triangle ADB = Angle C of the triangle ABC.

 

This means the triangle ADB is similar to the triangle ABC.

 

Correspondingly, triangle BDC and triangle ABC are similar.

 

Thus, the perpendicular BD of a right-angled triangle divides the Triangle ABC into two triangles which are similar to the parent triangle ABC.

 

Using this theorem, we can prove the Pythagoras theorem that is AB²+ BC²= AC².

 

We have already proved that the triangle ADB is similar to triangle ABC 

 

[frac{AD}{AB} = frac{AB}{AC}]

 

AD . AC = AB²

 

For the triangle, BDC and ABC

 

[frac{CD}{BC} = frac{BC}{AC}]

 

CD . AC = BC² 

 

Adding equation (i) and (ii) 

 

AD.AC + CD.AC = AB² + BC²

 

AC(AD + CD) = AB² + BC²

 

AC.AC = AB² + BC²

 

AC² = AB² + BC²

 

Hence, it is proved that the square of the hypotenuse is equal to the sum of the square of base and perpendicular of a right-angled triangle.

 

(Hypotenuse)² = (Base)² + (Perpendicular)²

 

Pythagorean Theorem Examples

Problem 1:

A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder. 

 

Solution:

Let AB be the ladder and CA
be the wall with the window at A.

 

Also, BC = 2.5 m and CA = 6 m.

 

From Pythagoras Theorem, we have:

AB² = BC² + CA²

 

AB² = (2.5)² + (6)²

 

AB² = 42.25 cm

 

AB = 6.5 cm

Therefore, the length of the ladder is 6.5 cm.

 

Problem 2:

A rectangle is of length 4 cm and diagonal of 5 cm. Find the perimeter of the rectangle.

 

Solution:

The angle between the two sides of a rectangle is 90 degrees. Thus, the diagonal half of a rectangle is the right-angled triangle where the diagonal of the rectangle is equal to the hypotenuse of a right-angled triangle, the length of the rectangle is the same as the perpendicular of the triangle and the breadth of the rectangle becomes the base of the triangle. 

 

For the right-angled triangle ABD,

 

AD² = AB² + BD²

5² = 4² + BD²

 

BD² = 5² – 4²

 

BD² = 25 – 16

 

BD² = 9

 

BD = 3

 

Therefore, the base of the right-angled triangle is 3 cm which is the breadth of the rectangle.

 

The perimeter of the rectangle = 2 x length + 2 x breadth

 

= (2 x 4) + (2 x 3)  = 8 + 6  = 14 cm.

 

The perimeter of the given rectangle is equal to 14 cm.

 

Application of Pythagorean Theorem

By applying the Pythagoras theorem, we can calculate the length of the sides of a right-angled triangle. Pythagoras theorem helps us to calculate the diagonal length of a roof, the height of a beam, the distance between the foot of the slanted bridge, and the perpendicular height. In architecture, we use the Pythagoras theorem to find the length of buildings, bridges, slopes. 

 

Also in wood construction, we use the theorem to find the length of different sides of the furniture. Pythagoras theorem is also used in Navigation as it is difficult to measure the distance in the sea or air. It can also be used to find the steepness of the slope of hills or mountains. 

Conclusion

This is the definition and proof of the Pythagorean theorem. Focus on how the theorem is proved using simple steps. Make sure you refer to the geometric images given so that you can understand the relation between the three sides of a right-angled triangle properly.