Question

Pythagorean triples are given by the formula x^2-y^2, 2xy, and x^2+y^2. Use the formulas for the Pythagorean triples to find a right triangle with leg lengths of 16 and an odd number. Show all of your work for full credit.

Answer 1

Let the unknown length be x

We have a formula to find the triplet of a right angle triangle

Put the given length equal to all three terms in the triplet

`2m,m^2-1,m^2+1`

put 16 equal to all above three but number comes out has to be integer.

`2m=16`

`\Rightarrow m=8`

Now,
`m^2-1=16`

`m^2=17` will not give integer.

And now,
`m^2+1=16`

`m^2=15` will not get integer value

Hence, the value of m will be 8

And substituting the value in the triplet formula we get

`2m=2\cdot 8=16` which is given.

`m^2-1=64-1=63` the other leg and is odd.

And
`m^2+1=64+1=65` the hypotenuse or the longest side.

Hence, 16,63 and 65 will form a right triangle

Pythagoras theorem is:

`\text{one side}^2+\text{other side}^2=hypotenuse^2`

Here, 16 and 63 are sides or legs and 65 is the hypotenuse.