Question

8
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 be a Pythagorean Triple for a right triangle that has a length of 180 and 181.

Answer 1

To find the missing values that complete the Pythagorean triples for the given right triangles, we can use the Pythagorean Theorem:

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

where \(c\) is the length of the hypotenuse and \(a\) and \(b\) are the lengths of the other two sides.

### Problem 1: Finding the missing value for lengths 20 and 99

Assuming that 99 is the hypotenuse (\(c\)), we can solve for the missing side (\(a\)):

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

Calculating this:

\[
400 + a^2 = 9801
\]

Now, isolate \(a^2\):

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

Now, take the square root to find \(a\):

\[
a = \sqrt{9401} \approx 97
\]

So, the missing value is approximately 97.

### Problem 2: Finding the missing value for lengths 180 and 181

Assuming that 181 is the hypotenuse (\(c\)), we can solve for the missing side (\(a\)):

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

Calculating this gives:

\[
32400 + a^2 = 32761
\]

Now, isolate \(a^2\):

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

Now, take the square root to find \(a\):

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

So, the missing value in the second case is 19.

### Summary:
1. For the sides 20 and 99, the missing value is approximately **97**.
2. For the sides 180 and 181, the missing value is **19**.