Question

A Pythagorean triple is a group of three whole numbers that satisfies the equation \(a^{2}+b^{2}=c^{2}\), where \(c\) is the measure of the hypotenuse. Some common Pythagorean triples are listed below. \(3,4,5 \quad 9,12,15 \quad 8,15,17 \quad 7,24,25\) a. List three other Pythagorean triples. b. Choose any whole number. Then multiply the whole number by each number of one of the Pythagorean triples you listed. Show that the result is also a Pythagorean triple.

Short answer

1. (5, 12, 13), (7, 24, 25), (8, 15, 17) 2. Triple (5, 12, 13) scaled by 2 gives (10, 24, 26), which is valid.

Step-by-step solution

List Three Pythagorean Triples

To find three different sets of Pythagorean triples, we can start by using the formula for generating Pythagorean triples: For two positive integers, \(m\) and \(n\), where \(m > n\), \(a = m^2 - n^2\), \(b = 2mn\), and \(c = m^2 + n^2\). Let's generate three sets with different \(m\) and \(n\) values:1. Let \(m = 2\) and \(n = 1\): - \(a = 2^2 - 1^2 = 3\), \(b = 2 \cdot 2 \cdot 1 = 4\), \(c = 2^2 + 1^2 = 5\), hence \((3, 4, 5)\). 2. Let \(m = 3\) and \(n = 2\): - \(a = 3^2 - 2^2 = 5\), \(b = 2 \cdot 3 \cdot 2 = 12\), \(c = 3^2 + 2^2 = 13\), hence \((5, 12, 13)\).3. Let \(m = 4\) and \(n = 3\): - \(a = 4^2 - 3^2 = 7\), \(b = 2 \cdot 4 \cdot 3 = 24\), \(c = 4^2 + 3^2 = 25\), hence \((7, 24, 25)\).

Select a Pythagorean Triple and Multiply

Choose one of the Pythagorean triples from Step 1, such as \((5, 12, 13)\), and multiply each number by a whole number, like 2.Multiply:- For \(5\): \(5 \cdot 2 = 10\)- For \(12\): \(12 \cdot 2 = 24\)- For \(13\): \(13 \cdot 2 = 26\)Thus, the new triple is \((10, 24, 26)\).

Verify the New Pythagorean Triple

Verify if \((10, 24, 26)\) forms a Pythagorean triple by checking if \(a^2 + b^2 = c^2 \).Calculate:- \(10^2 + 24^2 = 100 + 576 = 676\)- \(26^2 = 676\)Since both sides are equal, \((10, 24, 26)\) is a Pythagorean triple.

Key concepts

Pythagorean triples

A Pythagorean triple consists of three whole numbers, or integers, that satisfy the equation \(a^2 + b^2 = c^2\). This relationship is derived from the famous Pythagorean Theorem, which applies to right-angled triangles. In such triangles, \(c\) represents the length of the hypotenuse, the side opposite the right angle, while \(a\) and \(b\) are the lengths of the other two sides.

• Examples of common Pythagorean triples include \((3, 4, 5)\), \((5, 12, 13)\), and \((7, 24, 25)\).
• These triples share a distinctive feature: each element is an integer, and together they fit squarely into the equation \(a^2 + b^2 = c^2\).

Pythagorean triples can be generated using a simple formula for any two positive integers, \(m\) and \(n\), where \(m > n\). The formula is:

• \(a = m^2 - n^2\)
• \(b = 2mn\)
• \(c = m^2 + n^2\)

By substituting different values for \(m\) and \(n\), you can discover new Pythagorean triples. For example, setting \(m = 3\) and \(n = 2\) gives us the triple \((5, 12, 13)\).

integer solutions

When we talk about integer solutions in the context of the Pythagorean Theorem, we mean sets of whole numbers that satisfy the equation \(a^2 + b^2 = c^2\). In essence, each Pythagorean triple is an integer solution to this equation.

• These integer solutions represent the side lengths of a right triangle.
• Finding integer solutions is significant because it allows us to prove that such triangles exist with all sides of whole number lengths.

A notable property of integer solutions is that they can be scaled up. If you multiply each number in a Pythagorean triple by the same whole number, the result is also a Pythagorean triple. For instance, multiplying each part of the triple \((5, 12, 13)\) by 2 results in \((10, 24, 26)\), which is also an integer solution.
This trait highlights the inherent flexibility and expansiveness of Pythagorean triples, enabling us to create infinite sets of integer solutions from just one originating triple.

geometric proofs

Geometric proofs offer a visual means of verifying mathematical concepts like the Pythagorean Theorem. Essentially, a geometric proof uses shapes and measurements to demonstrate why a mathematical statement is true.

• For the Pythagorean Theorem, one simple geometric proof involves constructing squares on each side of a right triangle and comparing areas.
• This approach visually illustrates how the sum of areas on the legs \(a^2 + b^2\) equates to the area of the square on the hypotenuse \(c^2\).

By seeing these relationships laid out in a diagram, the logic underpinning the Pythagorean Theorem becomes clearer. The visual proof complements algebraic demonstrations and reinforces why the theorem consistently holds true for any right triangle regardless of its side lengths. This method embraces the effectiveness of a hands-on approach to understanding complex mathematical ideas.