Finding the Sum of All Possible Hypotenuses of a Right Triangle with Relatively Prime Legs

In the realm of right triangles, exploring the relationships between the lengths of the sides can lead to intriguing mathematical puzzles. Specifically, when the sides of a right triangle are relatively prime integers and the smallest leg is 28, the problem of finding the sum of all possible hypotenuses becomes both challenging and fascinating. Let's dive into the details and use some number theory to solve this puzzle.

Introduction to the Problem

A right triangle is characterized by its three sides: the two legs (A and B) and the hypotenuse (C). In this case, the smallest leg (A) is given as 28, and we aim to find all possible hypotenuses (C) which, along with the other sides, form a right triangle and are relatively prime integers. Relatively prime integers are those that have no common divisors other than 1, making this problem particularly interesting.

Identifying Possible Sides and Hypotenuses

One approach to solving this problem is by identifying possible pairs of legs that fit the criteria. Let's start with the given hint that three relatively prime integers above 28 are 29, 31, and 37. We can use these integers to form the hypotenuse (C) for our triangle, given that the smallest leg is 28.

Calculating Potential Hypotenuses

The formula to calculate the hypotenuse (C) for a right triangle with legs A and B is given by:

$$C sqrt{A^2 B^2}$$

Let's explore each of the given integers to determine if they can form the hypotenuse with the smallest leg of 28.

1. 29 as a Hypotenuse

$$C sqrt{28^2 29^2} approx 40.31$$ This value is not an integer, so 29 cannot be the hypotenuse.

2. 31 as a Hypotenuse

$$C sqrt{28^2 31^2} approx 41.77$$ This value is also not an integer, so 31 cannot be the hypotenuse.

3. 37 as a Hypotenuse

$$C sqrt{28^2 37^2} approx 46.40$$ This value is not an integer, so 37 cannot be the hypotenuse.

Given the results, these specific integers do not form valid right triangles with the smallest leg of 28. However, the problem suggests a more profound approach using number theory.

Using Number Theory for a Systematic Solution

Another method involves using the general formulas for Pythagorean triples. A right-angled triangle with relatively prime legs (A and B) and hypotenuse (C) can be generated using the formula:

$$B 2mn, quad A m^2 - n^2, quad C m^2 n^2$$

Where (m) and (n) are positive integers such that (m > n), (m - n) is odd, and gcd(m, n) 1. Given that the smallest leg (A 28), we can set up the equation:

$$28 m^2 - n^2$$

Solving for (m) and (n), we get the factorizations of 28 into the product of two relatively prime numbers. The factorizations are (14 times 1) and (7 times 2).

Exploring Factorizations

Let's use the factorizations to find possible hypotenuses.

1. Using (m 14), (n 1)

Substituting (m 14) and (n 1) into the formulas, we get:

$$A 14^2 - 1 195$$ $$B 2 times 14 times 1 28$$ $$C 14^2 1^2 197$$

So, one possible right triangle has sides 28, 195, and 197. Hence, the hypotenuse is 197.

2. Using (m 7), (n 2)

Substituting (m 7) and (n 2) into the formulas, we get:

$$A 7^2 - 2^2 45$$ $$B 2 times 7 times 2 28$$ $$C 7^2 2^2 53$$

So, another possible right triangle has sides 28, 45, and 53. Hence, the hypotenuse is 53.

Sum of All Possible Hypotenuses

Adding the two possible hypotenuses together:

$$197 53 250$$

Thus, the sum of all possible values of the hypotenuse is 250.

Conclusion: Through thorough exploration and application of number theory, we have determined that the sum of all possible hypotenuses for a right triangle with a smallest leg of 28 and relatively prime sides is 250.

Keywords: hypotenuse, right triangle, relatively prime integers