Pythagorean Theorem

Pythagoras Theorem

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

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 16cm 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 side 3 cm and 4 cm be square A and B respectively. Let the bigger square be C.

 

Step 2: Finding the area of square 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

 

Step 3: We can find the area of the third square by applying the given formula.

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

                                                                              

Step 4: Finding the length of the Square C.

 Area = \[{(side)^2}\]

Side = \[\sqrt {Area} \]

Side = \[\sqrt {25\,c{m^2}} \]                                                

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 as a Pythagorean triple. Pythagoras theorem questions involve the application of Pythagorean triple.

Geometrical Proof of Pythagorean Theorem 

State and Prove Pythagorean theorem

Pythagoras theorem class 10 CBSE contains geometrical proof of Pythagoras by using right-angled triangles. Pythagoras Theorem proof by using the concept of similarity of triangles is given below:

 

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

 

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 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 Pythagoras theorem that is AB2+ BC2= AC2.

 

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

 

AD/AB = AB/AC

AD . AC = AB2

 

For the triangle, BDC and ABC

CD/BC = 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 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 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 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 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. Suppose if we have two destinations, one at 400 km North of us and the other at 200 km East then we can actually find the distance between the two destinations by using Pythagoras theorem. Cartographer also finds the distance between two points by using Pythagoras theorem to represent it on a 2D sheet or map. It can also be used to find the steepness of the slope of hills or mountains.