Question
Jamar wants to generate a Pythagorean triple when x equals 12 and y equals 9 using the identity (x2+y2)2=(x2−y2)2+(2xy)2 (1 point) Responses 108 108 225 225 3969 3969 441
Answers
GPT-4o mini
To generate a Pythagorean triple using the identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\), we can start by calculating \(x^2\), \(y^2\), and other expressions.
1. Calculate \(x^2\) and \(y^2\):
\[
x = 12 \quad \Rightarrow \quad x^2 = 12^2 = 144
\]
\[
y = 9 \quad \Rightarrow \quad y^2 = 9^2 = 81
\]
2. Now, calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 144 + 81 = 225
\]
3. Calculate \((x^2 + y^2)^2\):
\[
(x^2 + y^2)^2 = 225^2 = 50625
\]
4. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 81 = 63
\]
5. Calculate \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 9 = 216
\]
6. Now, compute \((x^2 - y^2)^2\) and \((2xy)^2\):
\[
(x^2 - y^2)^2 = 63^2 = 3969
\]
\[
(2xy)^2 = 216^2 = 46656
\]
7. Finally, verify the identity:
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
\[
50625 = 3969 + 46656
\]
\[
50625 = 50625 \quad \text{(True)}
\]
Using the values of \(x = 12\) and \(y = 9\), the Pythagorean triple generated via this method includes the values \(63\) (from \(x^2 - y^2\)), \(216\) (from \(2xy\)), and \(225\) (from \(x^2 + y^2\)).
The numbers related to the computed Pythagorean triples include 225, 3969, and possibly other calculations. Among your listed responses, the result of interest from the identity is indeed:
**225**.
1. Calculate \(x^2\) and \(y^2\):
\[
x = 12 \quad \Rightarrow \quad x^2 = 12^2 = 144
\]
\[
y = 9 \quad \Rightarrow \quad y^2 = 9^2 = 81
\]
2. Now, calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 144 + 81 = 225
\]
3. Calculate \((x^2 + y^2)^2\):
\[
(x^2 + y^2)^2 = 225^2 = 50625
\]
4. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 81 = 63
\]
5. Calculate \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 9 = 216
\]
6. Now, compute \((x^2 - y^2)^2\) and \((2xy)^2\):
\[
(x^2 - y^2)^2 = 63^2 = 3969
\]
\[
(2xy)^2 = 216^2 = 46656
\]
7. Finally, verify the identity:
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
\[
50625 = 3969 + 46656
\]
\[
50625 = 50625 \quad \text{(True)}
\]
Using the values of \(x = 12\) and \(y = 9\), the Pythagorean triple generated via this method includes the values \(63\) (from \(x^2 - y^2\)), \(216\) (from \(2xy\)), and \(225\) (from \(x^2 + y^2\)).
The numbers related to the computed Pythagorean triples include 225, 3969, and possibly other calculations. Among your listed responses, the result of interest from the identity is indeed:
**225**.