Write a Pythagorean triplet whose smallest number is 14. One member of a Pythagorean triplet is 18. Find other two members of the triplet.

Find a Pythagorean triplet with smallest number 14: Let a = 14 , then find b and c such that 1 4 2 + b 2 = c 2 . The triplet is ( 14 , 48 , 50 ) .
Find a Pythagorean triplet with one member 18: Case 1: 18 is one of the shorter sides. Let a = 18 , then find b and c such that 1 8 2 + b 2 = c 2 . The triplets are ( 18 , 80 , 82 ) and ( 18 , 24 , 30 ) .
Case 2: 18 is the hypotenuse. There are no integer solutions in this case.
The Pythagorean triplet with smallest number 14 is ( 14 , 48 , 50 ) ​ , and the Pythagorean triplets with one member 18 are ( 18 , 24 , 30 ) ​ and ( 18 , 80 , 82 ) ​ .

Explanation

Problem Analysis We are given two problems:

Find a Pythagorean triplet whose smallest number is 14.

Find a Pythagorean triplet where one of the numbers is 18.

Definition of Pythagorean Triplet A Pythagorean triplet consists of three positive integers a , b , and c , such that a 2 + b 2 = c 2 .

Finding the Triplet with Smallest Number 14 For the first problem, we want to find a triplet where the smallest number is 14. Let a = 14 . Then we need to find integers b and c such that 1 4 2 + b 2 = c 2 . This can be rewritten as 196 + b 2 = c 2 , or 196 = c 2 − b 2 = ( c + b ) ( c − b ) . We need to find two factors of 196, say x and y , such that c + b = y and c − b = x . Then c = ( x + y ) /2 and b = ( y − x ) /2 . Since b and c must be integers, x and y must both be even or both be odd. Since 196 is even, both factors must be even. The even factor pairs of 196 are (2, 98) and (14, 14).


If c − b = 2 and c + b = 98 , then 2 c = 100 , so c = 50 , and 2 b = 96 , so b = 48 . Thus, the triplet is (14, 48, 50). Since 14 is the smallest number, this is a valid solution.
If c − b = 14 and c + b = 14 , then 2 c = 28 , so c = 14 , and 2 b = 0 , so b = 0 . This is not a valid solution since b must be a positive integer.

Finding the Triplet with One Member 18 For the second problem, we are given that one member of a Pythagorean triplet is 18. We consider two cases:

Case 1: 18 is one of the shorter sides. Let a = 18 . Then we need to find integers b and c such that 1 8 2 + b 2 = c 2 . This can be rewritten as 324 + b 2 = c 2 , or 324 = c 2 − b 2 = ( c + b ) ( c − b ) . We need to find two factors of 324, say x and y , such that c + b = y and c − b = x . Then c = ( x + y ) /2 and b = ( y − x ) /2 . Since b and c must be integers, x and y must both be even or both be odd. Since 324 is even, both factors must be even. The even factor pairs of 324 are (2, 162), (6, 54), (18, 18).
If c − b = 2 and c + b = 162 , then 2 c = 164 , so c = 82 , and 2 b = 160 , so b = 80 . Thus, the triplet is (18, 80, 82).
If c − b = 6 and c + b = 54 , then 2 c = 60 , so c = 30 , and 2 b = 48 , so b = 24 . Thus, the triplet is (18, 24, 30).
If c − b = 18 and c + b = 18 , then 2 c = 36 , so c = 18 , and 2 b = 0 , so b = 0 . This is not a valid solution since b must be a positive integer.
Case 2: 18 is the hypotenuse. Then we need to find integers a and b such that a 2 + b 2 = 1 8 2 = 324 . We can test integer values for a from 1 to 17 and see if b = 324 − a 2 ​ is an integer. However, there are no integer solutions in this case.

Final Answer Therefore, the Pythagorean triplet with smallest number 14 is (14, 48, 50), and the Pythagorean triplets with one member 18 are (18, 80, 82) and (18, 24, 30).

Examples
Pythagorean triplets are useful in construction and navigation. For example, if you are building a right-angled triangle structure and one side is 14 meters and another is 48 meters, you can use the Pythagorean theorem to find that the hypotenuse must be 50 meters to ensure a perfect right angle. Similarly, if you are navigating and know two sides of a right triangle, you can find the third side using Pythagorean triplets, which is essential for accurate positioning and course plotting.

Answered by GinnyAnswer | 2025-07-08