Type the correct answer in each box. in part e, you proved that any pythagorean triple can be generated using the identity (x^2 − y^2)^2 + (2xy)^2 = (x^2 + y^2)^2. find the missing x- and y-values and pythagorean triples using the identity given. write the triple in parentheses, without spaces between the values, with a comma between values, and in order from least to greatest.
x-value y-value pythagorean triple
4 3
5 (9,40,41)
6 3 (27,36,45)
7 5

(x²-y²)² + (2xy)² = (x²+y²)²

Find the missing x- and y-values and Pythagorean triples using the identity given

A Pythagorean triple consists of three positive integers a, b, and c, that satisfy the equation from the Pythagorean theorem, thus, a² + b² = c², such triple is commonly written (a,b,c).

We are given the equation : (x²-y²)² + (2xy)² = (x²+y²)² since this, we have :

a = (x²-y²)

b = (2xy)

c = (x²+y²)

Question 1)

X Value = 4

Y Value = 3

Pythagorean triples: ?

We can replace the values of x and y, to determine a, b and c.

a = (x²-y²) = (4²-3²) = 16-9 = 7

b = (2xy) = (2*4*3) = 24

c = (x²+y²) = (4²+3²) = 16+9 = 25

Answer 1 : Pythagorean triples : (7,24,25)

Question 2)

X Value = 5

Y Value = ?

Pythagorean Triples: (9,40,41)

Now we have a, b, and c, to determine Y

b = (2xy) = 40

Y = 40/2x = 40/2*5 = 40/10 = 4

Answer 2 : Y = 4

Question 3)

X Value = ?

Y Value = 3

Pythagorean Triples: (27,36,45)

Now we have a, b, and c, to determine X

b = (2xy) = 36

X = 36/2y = 36/2*3 = 36/6 = 6

Answer 3 : X = 6

Question 4)

X Value = 7

Y Value = 5

Pythagorean Triples: ?

We can replace the values of x and y, to determine a, b and c.

a = (x²-y²) = (7²-5²) = 49-25 = 24

b = (2xy) = (2*7*5) = 70

c = (x²+y²) = (7²+5²) = 49+25 = 74

Answer 4 : Pythagorean triples : (24,70,74)

Hope this helps!

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