One member of a Pythagorean triplet is 18. Find the other two members of the triplet.
Is 10, 24, and 26 a Pythagorean triplet?

Find the Pythagorean triplet containing 18 using the formulas a = m 2 − 1 , b = 2 m , and c = m 2 + 1 , where b = 18 . This gives the triplet (18, 80, 82).
Check if 10, 24, and 26 form a Pythagorean triplet by verifying if 1 0 2 + 2 4 2 = 2 6 2 .
Calculate 1 0 2 + 2 4 2 = 100 + 576 = 676 and 2 6 2 = 676 . Since they are equal, 10, 24, and 26 form a Pythagorean triplet.
The other two members of the Pythagorean triplet are 80 , 82 ​ , and 10, 24, and 26 is ​ a Pythagorean triplet.

Explanation

Problem Analysis We are given that one member of a Pythagorean triplet is 18. We need to find the other two members and also determine if 10, 24, and 26 form a Pythagorean triplet. A Pythagorean triplet consists of three positive integers a , b , and c , such that a 2 + b 2 = c 2 .

Finding the Pythagorean Triplet Let's find the Pythagorean triplet containing 18. We can consider two cases: either 18 is an even number and can be represented as 2 m , or it is part of the triplet such that a 2 + 1 8 2 = c 2 or 1 8 2 + b 2 = c 2 . Since 18 is even, we can express it as 2 m , where m = 9 . Then we can find a Pythagorean triplet using the formulas a = m 2 − 1 , b = 2 m , and c = m 2 + 1 . In our case, b = 18 , so m = 9 . Thus, a = 9 2 − 1 = 81 − 1 = 80 , and c = 9 2 + 1 = 81 + 1 = 82 . Therefore, one Pythagorean triplet is (18, 80, 82).

Checking the Pythagorean Triplet Now, let's check if 10, 24, and 26 form a Pythagorean triplet. We need to verify if 1 0 2 + 2 4 2 = 2 6 2 . We have 1 0 2 = 100 and 2 4 2 = 576 . So, 1 0 2 + 2 4 2 = 100 + 576 = 676 . Also, 2 6 2 = 676 . Since 1 0 2 + 2 4 2 = 2 6 2 , the numbers 10, 24, and 26 form a Pythagorean triplet.

Conclusion Therefore, the other two members of the Pythagorean triplet containing 18 are 80 and 82. And 10, 24, and 26 do form a Pythagorean triplet.


Examples
Pythagorean triplets are useful in construction and navigation. For example, when building a right-angled structure, builders can use a Pythagorean triplet to ensure that the angle is exactly 90 degrees. A simple example is using a 3-4-5 triplet to create a right triangle. Similarly, in navigation, Pythagorean triplets can be used to calculate distances and directions, especially when dealing with right triangles on maps or in space.

Answered by GinnyAnswer | 2025-07-08