Finding Pythagorean Triples with a Hypotenuse of 85 Using Paper and Pencil

Pythagorean triples are sets of three positive integers (a), (b), and (c) that satisfy the Pythagorean theorem, (a^2 b^2 c^2). If the hypotenuse (c) is 85, finding the corresponding integer values of (a) and (b) can be done using paper and pencil with a series of calculations and checks for perfect squares.

Understanding Pythagorean Triples

Using the Pythagorean theorem, we can find the right triangle sides that form a Pythagorean triple with a hypotenuse of 85. The equation is:

a^2 b^2 85^2

Steps to Find the Pythagorean Triples

1. Calculate the value of (c^2):
85^2 7225

2. Set up the equation: a^2 b^2 7225

3. Loop through possible values of (a) from 1 to 84 (since (a) must be less than the hypotenuse (c)). For each value of (a), calculate (b^2):
b^2 7225 - a^2

4. Check if (b^2) is a perfect square. If it is, calculate (b) as the square root of (b^2).

Example Calculations

Here are some example calculations to find the Pythagorean triples:

For (a 7)

b^2 7225 - 7^2 7225 - 49 7176
7176 is not a perfect square.

For (a 56)

b^2 7225 - 56^2 7225 - 3136 4089
4089 is not a perfect square.

For (a 60)

b^2 7225 - 60^2 7225 - 3600 3625
3625 is not a perfect square.

For (a 65)

b^2 7225 - 65^2 7225 - 4225 3000
3000 is not a perfect square.

For (a 75)

b^2 7225 - 75^2 7225 - 5625 1600
b sqrt{1600} 40
Thus, one Pythagorean triple is 75, 40, 85

For (a 84)

b^2 7225 - 84^2 7225 - 7056 169
b sqrt{169} 13
Thus, another Pythagorean triple is 84, 13, 85

Summary of Pythagorean Triples with Hypotenuse 85

The Pythagorean triples with a hypotenuse of 85 are:

75, 40, 85 84, 13, 85

These are the two Pythagorean triples with a hypotenuse of 85.

Alternative Methods

Another approach involves using the properties of Pythagorean triples generated by (m) and (n), where (m) and (n) are positive integers, and m^2 - n^2, 2mn, m^2 n^2 form a Pythagorean triple. If (m 9) and (n 2), then 81 - 4 85, and the triple is 36, 77, 85. If (m 7) and (n 6), then 49 - 36 85, and the triple is 13, 84, 85.

Example Using ((x, y))

Let x be the base and y be the perpendicular of a triangle with hypotenuse 85, then:

85^2 x^2 y^2

85^2 - x^2 y^2

85 - x sqrt{85^2 - x^2}

We need to choose (x) in such a way that it makes the left-hand side a perfect square.

For x 84. Left hand side 1 * 169 13^2

For x 77. Left hand side 8 * 162 16 * 81 36^2

For x 75. Left hand side 10 * 160 1600 40^2

For x 68. Left hand side 17 * 153 17^2 * 3^2 51^2

Therefore, we have the following Pythagorean triples:

85, 84, 13 85, 77, 36 85, 75, 40 85, 68, 51