Question

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

Answer 1

Answer: (9, 40, 41)

Explanation:

There are an infinite number of Pythagorean triples.

Any value a, b, c such that a² + b² = c²

Here are a few of them:

3, 4, 5

5, 12, 13

7, 24, 25

8, 15, 17

9, 40, 41

Answer 2

​(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!​​​​​

Explanation:

From Spymore