A pythagorean triple consists of three positive integers that work with the pythagorean theorem. Find the missing value that be a pythagorean triple for a right triangle that has a length of 20 and 99. Find the missing value that would be a pythagorean triple for a right triangle that has length of 180 and 181.

Answer 1

To find the missing value that forms a Pythagorean triple for each of the specified triangles, we can use the Pythagorean theorem, which states:

\[ a^2 + b^2 = c^2 \]

where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs of the right triangle.

Problem 1: Given lengths of 20 and 99

Let's assume that 99 is the length of the hypotenuse (\(c\)) and 20 is one of the legs (\(a\)). We need to find the missing leg (\(b\)).

Using the Pythagorean theorem:

\[ 20^2 + b^2 = 99^2 \]

Calculating \(20^2\) and \(99^2\):

\[ 400 + b^2 = 9801 \]

Now, isolate \(b^2\):

\[ b^2 = 9801 - 400 \] \[ b^2 = 9401 \]

Now, taking the square root to find \(b\):

\[ b = \sqrt{9401} \]

Calculating \(b\):

\[ b \approx 96.96 \]

Since we need \(b\) to be a positive integer, we won't have an integer Pythagorean triple with 20 and 99 as one of the sides.

Problem 2: Given lengths of 180 and 181

Now let's find the missing value (leg \(b\)) when one leg \(a = 180\) and the hypotenuse \(c = 181\).

Using the Pythagorean theorem:

\[ 180^2 + b^2 = 181^2 \]

Calculating \(180^2\) and \(181^2\):

\[ 32400 + b^2 = 32761 \]

Now, isolate \(b^2\):

\[ b^2 = 32761 - 32400 \] \[ b^2 = 361 \]

Taking the square root to find \(b\):

\[ b = \sqrt{361} = 19 \]

Thus, for the right triangle with sides 180, 19, and 181, we have a valid Pythagorean triple.

Summary

• For the first triangle (20 and 99), no integer \(b\) exists such that it forms a Pythagorean triple.
• For the second triangle (180 and 181), the missing value is \(b = 19\).

Final answer:

1. No integer value for the first triangle.
2. Missing value for the second triangle is \(19\).