What is the form of the Pythagorean triple generator?
A. [tex]$\left(x^2+y^2\right)^2=\left(x^2+y^2\right)^2+(2 x y)^2$[/tex]
B. [tex]$\left(x^2-y^2\right)^2=\left(x^2+y^2\right)^2-(2 x y)^2$[/tex]
C. [tex]$\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2 x y)^2$[/tex]
D. [tex]$\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2-(2 x y)^2$[/tex]
Answer (2)
The Pythagorean triple generator can be derived using the relationships a = x 2 − y 2 , b = 2 x y , and c = x 2 + y 2 . Among the options given, Option C correctly represents this relationship as ( x 2 + y 2 ) 2 = ( x 2 − y 2 ) 2 + ( 2 x y ) 2 . Therefore, the correct answer is Option C.
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The Pythagorean theorem states a 2 + b 2 = c 2 .
A Pythagorean triple can be generated by a = x 2 − y 2 , b = 2 x y , and c = x 2 + y 2 .
Substitute these expressions into the Pythagorean theorem: ( x 2 − y 2 ) 2 + ( 2 x y ) 2 = ( x 2 + y 2 ) 2 .
The correct equation representing the Pythagorean triple generator is C .
Explanation
Problem Analysis and Pythagorean Theorem We are asked to identify the correct form of the Pythagorean triple generator from the given options. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as a 2 + b 2 = c 2 , where a and b are the lengths of the legs, and c is the length of the hypotenuse. A Pythagorean triple consists of three positive integers a , b , and c that satisfy this equation. The Pythagorean triple generator expresses a , b , and c in terms of two integers x and y . A common form is a = x 2 − y 2 , b = 2 x y , and c = x 2 + y 2 . We need to verify which of the given options correctly represents this relationship.
Analyzing Option A Let's analyze option A: ( x 2 + y 2 ) 2 = ( x 2 + y 2 ) 2 + ( 2 x y ) 2 . This equation implies that ( 2 x y ) 2 = 0 , which means either x = 0 or y = 0 . This is not a general form for generating Pythagorean triples, so option A is incorrect.
Analyzing Option B Now, let's analyze option B: ( x 2 − y 2 ) 2 = ( x 2 + y 2 ) 2 − ( 2 x y ) 2 . Rearranging this equation, we get ( x 2 − y 2 ) 2 + ( 2 x y ) 2 = ( x 2 + y 2 ) 2 . Expanding the left side, we have ( x 4 − 2 x 2 y 2 + y 4 ) + 4 x 2 y 2 = x 4 + 2 x 2 y 2 + y 4 . Expanding the right side, we have ( x 2 + y 2 ) 2 = x 4 + 2 x 2 y 2 + y 4 . Since both sides are equal, this equation is a valid representation of the Pythagorean theorem.
Analyzing Option C Next, let's analyze option C: ( x 2 + y 2 ) 2 = ( x 2 − y 2 ) 2 + ( 2 x y ) 2 . Expanding the right side, we have ( x 4 − 2 x 2 y 2 + y 4 ) + 4 x 2 y 2 = x 4 + 2 x 2 y 2 + y 4 . The left side is ( x 2 + y 2 ) 2 = x 4 + 2 x 2 y 2 + y 4 . Since both sides are equal, this equation is also a valid representation of the Pythagorean theorem.
Analyzing Option D Finally, let's analyze option D: ( x 2 + y 2 ) 2 = ( x 2 − y 2 ) 2 − ( 2 x y ) 2 . Rearranging this equation, we get ( x 2 + y 2 ) 2 + ( 2 x y ) 2 = ( x 2 − y 2 ) 2 . This is not a valid representation of the Pythagorean theorem because it implies that the sum of two squares is equal to another square, which is not generally true for Pythagorean triples.
Determining the Correct Option Both options B and C are valid representations of the Pythagorean theorem. However, the standard form of the Pythagorean triple generator is given by a = x 2 − y 2 , b = 2 x y , and c = x 2 + y 2 , which corresponds to the equation ( x 2 + y 2 ) 2 = ( x 2 − y 2 ) 2 + ( 2 x y ) 2 . Therefore, option C is the correct answer.
Final Answer The correct form of the Pythagorean triple generator is C .
Examples
Pythagorean triples are useful in various fields, such as construction and navigation. For example, a construction worker might use a 3-4-5 right triangle to ensure that a corner is perfectly square. Similarly, navigators can use Pythagorean triples to calculate distances and bearings. The Pythagorean triple generator allows us to find sets of integers that satisfy the Pythagorean theorem, making these calculations easier.
