HOW TO FIND PYTHAGOREAN TRIPLET FROM ONE NUMBER

Example 1 :

Find the Pythagorean triplet in which one number is 8.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m² - 1), (m² + 1).

Let us consider the given number as 2m

2m  =  8

m  =  4

The triplet is 8, 15 and 17.

Verifying the relationship :

17²  =  15² + 8²

289  =  225 + 64

289  =  289

Example 2 :

Find the Pythagorean triplet in which one number is 12.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m² - 1), (m² + 1).

Let us consider the given number as 2m

2m  =  12

m  =  6

The triplet is 12, 35 and 37.

Verifying the relationship :

37²  =  35² + 12²

1369  =  1225 + 144

1369  =  1369

Example 3 :

Find the Pythagorean triplet in which one number is 14.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m² - 1), (m² + 1).

Let us consider the given number as 2m

2m  =  14

m  =  7

The triplet is 14, 48 and 50.

Verifying the relationship :

50²  =  48² + 14²

2500  =  2304 + 196

2500  =  2500

Example 4 :

Find the Pythagorean triplet in which one number is 6.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m² - 1), (m² + 1).

Let us consider the given number as 2m

2m  =  6

m  =  3

The triplet is 6, 8 and 10.

Verifying the relationship :

10²  =  8² + 6²

100  =  64 + 36

100  =  100

Example 5 :

Find the Pythagorean triplet in which one number is 16.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m² - 1), (m² + 1).

Let us consider the given number as 2m

2m  =  16

m  =  8

The triplet is 16, 63 and 65.

Verifying the relationship :

65²  =  63² + 16²

4225  =  3969 + 256

4225  =  4225

Example 5 :

Suppose m and n are two numbers. If m² - n², 2mn and m² + n² are the three sides of a triangle, then show that it is  a right triangle and hence write any two pairs of Pythagorean triplet.

Solution :

The sides of the right triangle are m² - n², 2mn and m² + n²

(m² - n²)² + (2mn)² = (m² + n²)²

L.H.S :

= (m² - n²)² + (2mn)²

= (m²)² - 2m² n² + (n²)² + 4m² n²

= (m²)² + 2m² n² + (n²)²

= (m² + n²)²

R.H.S

So, the given measures are sides of the right triangle.

Example 6 :

In a right angled triangle if one side forming the right angle is 6 and the hypotenuse is 10. What is the length of the other right angle forming side?

a) 8       b) 10        c) 12      d) 6

Solution :

The longest side of a right triangle = hypotenuse

Let the other side be x.

The other two sides are 6 and x.

Square of hypotenuse = sum of squares of remaining two sides

10² = 6² + x²

100 = 36 + x²

x² = 100 - 36

x² = 64

x = 8

So, the other side of the right triangle = 8 cm.

Option a is correct.

Example 6 :

Pythagoras theorem can only be applied on ________ triangles.

a) equilateral    b) isosceles    c) right angled

d) isosceles right angled

Solution :

Pythagoras theorem can only be applied on right angled triangles.

Example 7 :

3, 4 & 5 are not a Pythagorean triplet.

a) True     b) False

Solution :

Let a = 3, b = 4 and c = 5

c² = a² + b²

5² = 3² + 4²

25 = 9 + 16

25 = 25

So, the given measures are Pythagorean triples. It is true.

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