Question
Find the missing value that be Pythagorean triple for a right triangle that had a length of 20 and 99
Answers
GPT-4o mini
To determine if there is a missing value \( c \) that forms a Pythagorean triple with the given values (20 and 99), we can use the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
Given two sides of the triangle, we can assume \( a = 20 \) and \( b = 99 \) (the lengths of the two legs). We want to find \( c \), which represents the length of the hypotenuse.
Calculating \( c \):
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 20^2 + 99^2
\]
\[
c^2 = 400 + 9801
\]
\[
c^2 = 10201
\]
\[
c = \sqrt{10201} = 101
\]
Thus, the missing value that forms a Pythagorean triple with the lengths 20 and 99 is \( c = 101 \). The complete Pythagorean triple is \( (20, 99, 101) \).
\[
a^2 + b^2 = c^2
\]
Given two sides of the triangle, we can assume \( a = 20 \) and \( b = 99 \) (the lengths of the two legs). We want to find \( c \), which represents the length of the hypotenuse.
Calculating \( c \):
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 20^2 + 99^2
\]
\[
c^2 = 400 + 9801
\]
\[
c^2 = 10201
\]
\[
c = \sqrt{10201} = 101
\]
Thus, the missing value that forms a Pythagorean triple with the lengths 20 and 99 is \( c = 101 \). The complete Pythagorean triple is \( (20, 99, 101) \).
Find the missing value that be a Pythagorean triple for a right triangle that had a length of 180 and 181
GPT-4o mini
To find the missing value that forms a Pythagorean triple with the lengths 180 and 181, we can again use the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
We need to determine if either 180 or 181 can be the hypotenuse \( c \) or one of the legs \( a \) or \( b \).
### Case 1: Assume \( c = 181 \)
If 181 is the hypotenuse, then we can find the missing leg \( b \) using:
\[
180^2 + b^2 = 181^2
\]
Calculating the squares:
\[
180^2 = 32400
\]
\[
181^2 = 32761
\]
Now substituting these values into the equation:
\[
32400 + b^2 = 32761
\]
\[
b^2 = 32761 - 32400
\]
\[
b^2 = 361
\]
\[
b = \sqrt{361} = 19
\]
So, one possible Pythagorean triple is \( (180, 19, 181) \).
### Case 2: Assume \( c = 180 \)
This case is invalid because \( 181 \) cannot be a leg if \( 180 \) is assumed to be the hypotenuse.
### Conclusion
The missing value that forms a Pythagorean triple with the lengths 180 and 181 is \( 19 \). Thus, the complete Pythagorean triple is \( (19, 180, 181) \).
\[
a^2 + b^2 = c^2
\]
We need to determine if either 180 or 181 can be the hypotenuse \( c \) or one of the legs \( a \) or \( b \).
### Case 1: Assume \( c = 181 \)
If 181 is the hypotenuse, then we can find the missing leg \( b \) using:
\[
180^2 + b^2 = 181^2
\]
Calculating the squares:
\[
180^2 = 32400
\]
\[
181^2 = 32761
\]
Now substituting these values into the equation:
\[
32400 + b^2 = 32761
\]
\[
b^2 = 32761 - 32400
\]
\[
b^2 = 361
\]
\[
b = \sqrt{361} = 19
\]
So, one possible Pythagorean triple is \( (180, 19, 181) \).
### Case 2: Assume \( c = 180 \)
This case is invalid because \( 181 \) cannot be a leg if \( 180 \) is assumed to be the hypotenuse.
### Conclusion
The missing value that forms a Pythagorean triple with the lengths 180 and 181 is \( 19 \). Thus, the complete Pythagorean triple is \( (19, 180, 181) \).