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:
- For the sides 20 and 99, the missing value is approximately 97.
- For the sides 180 and 181, the missing value is 19.