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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________
Example 18: Find the Pythagorean triplet whose smallest member is 10.
2
2
Solution: We can obtain a Pythagorean triplet by using the general form 2m, m – 1, m + 1.
2
2
Let us first take m – 1 = 10 ⇒ m = 11
But 11 is not a perfect square, so m will not be a natural number.
Let us consider 2m = 10
All pythagorean triplets
2
⇒ m = 5 ⇒ m – 1 = 25 – 1 = 24 Note: may not be obtained using
2
and m + 1 = 25 + 1 = 26 this form. For example, 5,
12, 13.
Thus, 10, 24 and 26 is the triplet with 10 as the smallest number.
Example 19: Find the other two numbers of a Pythagorean triplet if one number is 14.
2
Solution: The general form of a Pythagorean triplet is (2m), (m – 1) and (m + 1).
2
2
The smallest number can be either m – 1 or 2m.
Let 2m = 14 ⇒ m = 7
\ m – 1 = 49 – 1 = 48
2
2
And m + 1 = 49 + 1 = 50
Therefore, the other two numbers of a Pythagorean triplet are 48 and 50.
Example 20: Find the other two numbers of a Pythagorean triplet if the smallest number is 9.
Solution: Here, the smallest number is 9, an odd number.
Let m = 9. Then, the other two numbers are:
2
2
m − 1 9 − 1 81 − 1
= = = 40
2 2 2 Note: If m is an odd and smallest number,
2
2
m + 1 9 + 1 81 + 1 then the Pythagorean triplet is of
2
2
and = = = 41 m − 1 m + 1
2 2 2 the form m, 2 and 2 .
Thus, the triplet is thus 9, 40 and 41.
Practice Time 5B
1. What will be the ones digit of the square of the following numbers?
(a) 82 (b) 93 (c) 128 (d) 179
2. By observation, tell whether the following numbers are perfect squares or not. Give reasons.
(a) 9057 (b) 33453 (c) 9928 (d) 34532
3. The square of which of the following numbers would be an odd number?
(a) 61 (b) 872 (c) 499 (d) 176
4. Without doing actual multiplication, find the square of
(a) 35 (b) 115 (c) 26 (d) 92
5. Write a Pythagorean triplet whose one member is
(a) 8 (b) 14 (c) 18
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