Answer :
To generate a Pythagorean triple using the Pythagorean Identity with [tex]\( x = 7 \)[/tex] and [tex]\( y = 4 \)[/tex], follow these steps:
1. Understand the Pythagorean Identity: For any right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse ([tex]\( z \)[/tex]) is equal to the sum of the squares of the lengths of the other two sides ([tex]\( x \)[/tex] and [tex]\( y \)[/tex]). This is expressed as:
[tex]\[
x^2 + y^2 = z^2
\][/tex]
2. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = 7^2 = 49
\][/tex]
3. Calculate [tex]\( y^2 \)[/tex]:
[tex]\[
y^2 = 4^2 = 16
\][/tex]
4. Find [tex]\( z^2 \)[/tex] by adding [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]:
[tex]\[
z^2 = 49 + 16 = 65
\][/tex]
5. Determine the values: Therefore, the Pythagorean triple is composed of [tex]\( x^2 = 49 \)[/tex], [tex]\( y^2 = 16 \)[/tex], and [tex]\( z^2 = 65 \)[/tex].
With these calculations, we find a Pythagorean triple that corresponds to option d) 33, 56, 65. Although in reduced form and integer values, option [tex]\( d \)[/tex] provides a scenario where the triple aligned with the calculations follows the pattern of a recognizable Pythagorean triple when solved for fundamental integer values.
These results confirm that 65 could potentially be the hypotenuse when [tex]\( x \)[/tex] and [tex]\( y \)[/tex] reflect squared values or scale to integer multiples which fit the conditions of the problem.
In conclusion, the correct answer based on the squared results and their relationships as described fits d) 33, 56, 65.
1. Understand the Pythagorean Identity: For any right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse ([tex]\( z \)[/tex]) is equal to the sum of the squares of the lengths of the other two sides ([tex]\( x \)[/tex] and [tex]\( y \)[/tex]). This is expressed as:
[tex]\[
x^2 + y^2 = z^2
\][/tex]
2. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = 7^2 = 49
\][/tex]
3. Calculate [tex]\( y^2 \)[/tex]:
[tex]\[
y^2 = 4^2 = 16
\][/tex]
4. Find [tex]\( z^2 \)[/tex] by adding [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]:
[tex]\[
z^2 = 49 + 16 = 65
\][/tex]
5. Determine the values: Therefore, the Pythagorean triple is composed of [tex]\( x^2 = 49 \)[/tex], [tex]\( y^2 = 16 \)[/tex], and [tex]\( z^2 = 65 \)[/tex].
With these calculations, we find a Pythagorean triple that corresponds to option d) 33, 56, 65. Although in reduced form and integer values, option [tex]\( d \)[/tex] provides a scenario where the triple aligned with the calculations follows the pattern of a recognizable Pythagorean triple when solved for fundamental integer values.
These results confirm that 65 could potentially be the hypotenuse when [tex]\( x \)[/tex] and [tex]\( y \)[/tex] reflect squared values or scale to integer multiples which fit the conditions of the problem.
In conclusion, the correct answer based on the squared results and their relationships as described fits d) 33, 56, 65.
