Question
Find the missing value that be a Pythagorean triple for a right triangle that has a length of 180 and 181.
Answers
GPT-4o mini
To find the missing value that can form a Pythagorean triple with the given lengths of \( 180 \) and \( 181 \), we can use the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
where \( c \) is the hypotenuse (the longest side), and \( a \) and \( b \) are the other two sides. First, we need to identify the roles of the given sides. Given that \( 181 \) is the larger number, we can assume \( 181 \) is the hypotenuse. The two sides are then \( 180 \) and the missing side, which we’ll call \( x \).
Applying the Pythagorean theorem, we have:
\[
180^2 + x^2 = 181^2
\]
Calculating \( 180^2 \) and \( 181^2 \):
\[
180^2 = 32400
\]
\[
181^2 = 32761
\]
Substituting these values into the equation gives:
\[
32400 + x^2 = 32761
\]
Now, to isolate \( x^2 \), we subtract \( 32400 \) from both sides:
\[
x^2 = 32761 - 32400
\]
\[
x^2 = 361
\]
Taking the square root of both sides, we find:
\[
x = \sqrt{361} = 19
\]
Thus, the missing value that completes the Pythagorean triple with \( 180 \) and \( 181 \) is \( \boxed{19} \).
\[
a^2 + b^2 = c^2
\]
where \( c \) is the hypotenuse (the longest side), and \( a \) and \( b \) are the other two sides. First, we need to identify the roles of the given sides. Given that \( 181 \) is the larger number, we can assume \( 181 \) is the hypotenuse. The two sides are then \( 180 \) and the missing side, which we’ll call \( x \).
Applying the Pythagorean theorem, we have:
\[
180^2 + x^2 = 181^2
\]
Calculating \( 180^2 \) and \( 181^2 \):
\[
180^2 = 32400
\]
\[
181^2 = 32761
\]
Substituting these values into the equation gives:
\[
32400 + x^2 = 32761
\]
Now, to isolate \( x^2 \), we subtract \( 32400 \) from both sides:
\[
x^2 = 32761 - 32400
\]
\[
x^2 = 361
\]
Taking the square root of both sides, we find:
\[
x = \sqrt{361} = 19
\]
Thus, the missing value that completes the Pythagorean triple with \( 180 \) and \( 181 \) is \( \boxed{19} \).