To generate a Pythagorean triple using the identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\), we need to substitute \(x = 12\) and \(y = 9\) into the formula.
First, we calculate \(x^2\) and \(y^2\):
\[ x^2 = 12^2 = 144 \] \[ y^2 = 9^2 = 81 \]
Now, we can compute \(x^2 + y^2\) and \(x^2 - y^2\):
\[ x^2 + y^2 = 144 + 81 = 225 \] \[ x^2 - y^2 = 144 - 81 = 63 \]
Next, we calculate \(2xy\):
\[ 2xy = 2 \times 12 \times 9 = 216 \]
Now we will square the values:
- \((x^2 + y^2)^2\):
\[ (225)^2 = 50625 \]
- \((x^2 - y^2)^2\):
\[ (63)^2 = 3969 \]
- \((2xy)^2\):
\[ (216)^2 = 46656 \]
Now, we check the identity:
\[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \]
Plugging in the values we calculated:
\[ 50625 = 3969 + 46656 \]
Calculating the right-hand side:
\[ 3969 + 46656 = 50625 \]
The identity holds, and now we look at the options provided:
- 108
- 225
- 3969
- 441
Among the options, \(3969\) is the result we calculated for \((x^2 - y^2)^2\). Thus, the correct answer from the options is:
3969
1 answer