Introduction:

Pythagoras was a Greek mathematician and philosopher. He was born in 570 BC, on the island of Samos. One of the famous relics of Pythagoras is the Pythagoras’ Theorem; after his name, which states "In a right triangle it is true that the square of the hypotenuse is equal to the sum of the squares of the other two sides". The hypotenuse is the longest side of a right triangle.

There are various methods of proving this theorem. We shall prove it by using similar triangles.


Statement:

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Figure:





Given:


1. ∆ACB is a right-angled triangle. Figure (i)
2. Where m∠C = 900 and m = a, m = b and m = c.

To Prove:

c² = a² + b²

Construction:

1. Draw a perpendicular from C on.
2. m = h, m = x and m = y.
3. c = x + y
4. The ∆ABC is now split into two right-angled triangles ∆ADC and ∆BDC as shown in figures (i)-a and (i)-b respectively.

Proof:

Statements

Reasons

In ∆ADC  ↔  ∆ACB

 

∠A ≅ ∠A

 

∠ADC ≅ ∠ACB

 


∠C ≅ ∠A

 

 

 


∴ ∆ADC ∼ ∆ACB

 

`\ frac {x}{b}` = `\ frac {b}{c}`

 

or

 

x =  `\ frac {b^2}{c}`  ---------(a)

 

 

Now in ∆BDC  ↔  ∆BCA

 

∠B ≅ ∠B

 

∠BDC ≅ ∠BCA

 


∠C ≅ ∠B

 

 

 


∴ ∆ADC ∼ ∆ACB

 

`\ frac {y}{a}` = `\ frac {a}{c}`

 

or

 

y =  `\ frac {a^2}{c}`  ---------(b)

 

 

x + y = `\ frac {b^2}{c}` + `\ frac {a^2}{c}`   

 

x + y = `frac {1}{c}` (b² +  a²)

 

or

 

c (x + y) = a² +  b²

 


c² = a² +  b²

 

Reference figures (i)-a and (i)

 

Same angle (common angle)

 

Each angle = 900 (Construction – given)

 

Congruency of the third angle of similar triangles.

(when two angles of a similar triangle are congruent the third angle is also congruent)

 

By Congruency of three angles

 

Proportional part of similar triangles

 

 

 

 

 

 

 

Reference figure (i)-b and (i)

 

Same angles (common angle)

 

Each angle = 900 (Construction – given)

 

Congruency of the third angle of similar triangles.

(when two angles of a similar triangle are congruent the third angle is also congruent)

 

By Congruency of three angles

 

Proportional part of similar triangles

 

 

 

 

 

 

 

By adding (a) and  (b)

 

 

 

Multiplying both sides by c

 

 


x + y = c   construction figure (i)

 

Hence the proof.


Applications:

There may be many applications of the Pythagorean Theorem in everyday life; some of them are as follow

1. The most common use of Pythagoras' Theorem is to find the lengths of sides of a right-angled triangle only!

2. It is used to find the lengths of the diagonal of a square, rectangles, etc.

3. It can be used to find trigonometric ratios like sine, cosine, tangent, etc.

4. The builders also use it to measure different lengths in their construction of a building. etc.


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