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98 MATHEMATICS

Can you find more such triplets?
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For any natural number m > 1, we have (2m) + (m – 1) = (m + 1) . So, 2m,
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m – 1 and m + 1 forms a Pythagorean triplet.
Try to find some more Pythagorean triplets using this form.
Example 2: Write a Pythagorean triplet whose smallest member is 8.
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Solution: We can get Pythagorean triplets by using general form 2m, m – 1, m + 1.
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Let us first take m – 1 = 8
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So, m = 8 + 1 = 9
which gives m = 3
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Therefore, 2m = 6 and m + 1 = 10
The triplet is thus 6, 8, 10. But 8 is not the smallest member of this.
So, let us try 2m = 8
then m = 4
We get m – 1 = 16 – 1 = 15
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and m + 1 = 16 + 1 = 17
The triplet is 8, 15, 17 with 8 as the smallest member.

Example 3: Find a Pythagorean triplet in which one member is 12.

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Solution: If we take m – 1 = 12
Then, m = 12 + 1 = 13
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Then the value of m will not be an integer.
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So, we try to take m + 1 = 12. Again m = 11 will not give an integer value for m.
So, let us take 2m = 12
then m = 6
Thus, m – 1 = 36 – 1 = 35 and m + 1 = 36 + 1 = 37
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Therefore, the required triplet is 12, 35, 37.
Note: All Pythagorean triplets may not be obtained using this form. For example another
triplet 5, 12, 13 also has 12 as a member.

EXERCISE 6.2
1. Find the square of the following numbers.
(i) 32 (ii) 35 (iii) 86 (iv) 93
(v) 71 (vi) 46
2. Write a Pythagorean triplet whose one member is.
(i) 6 (ii) 14 (iii) 16 (iv) 18

6.5 Square Roots
Study the following situations.
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(a) Area of a square is 144 cm . What could be the side of the square?