1.        Find a Pythagorean triplet whose smallest member is 8.

To find a Pythagorean triplet where the smallest member is \( 8 \), we use the standard form for generating Pythagorean triplets, which are integers \( (a, b, c) \) that satisfy:

\[
a^2 + b^2 = c^2
\]

There are two approaches to finding Pythagorean triplets:

### Method 1: Using the formula for generating Pythagorean triplets
For any pair of integers \( m \) and \( n \) where \( m > n \), the triplet \( (a, b, c) \) is given by:

\[
a = m^2 - n^2
\]
\[
b = 2mn
\]
\[
c = m^2 + n^2
\]

We want \( a = 8 \), so let's solve for \( m \) and \( n \):

\[
m^2 - n^2 = 8
\]

This simplifies to:

\[
(m - n)(m + n) = 8
\]

Now, find factors of 8:

- \( m - n = 1 \) and \( m + n = 8 \)

Solving this system of equations:

1. \( m - n = 1 \)
2. \( m + n = 8 \)

Add these two equations:

\[
2m = 9 \implies m = \frac{9}{2}
\]

This doesn't give an integer value, so we move to the next pair of factors.

- \( m - n = 2 \) and \( m + n = 4 \)

Solving this system:

1. \( m - n = 2 \)
2. \( m + n = 4 \)

Adding these:

\[
2m = 6 \implies m = 3
\]

Substitute \( m = 3 \) into \( m + n = 4 \):

\[
3 + n = 4 \implies n = 1
\]

### Step 2: Calculate \( b \) and \( c \)

Using \( m = 3 \) and \( n = 1 \):

\[
a = m^2 - n^2 = 3^2 - 1^2 = 9 - 1 = 8
\]
\[
b = 2mn = 2(3)(1) = 6
\]
\[
c = m^2 + n^2 = 3^2 + 1^2 = 9 + 1 = 10
\]

### The Pythagorean triplet is \( (8, 6, 10) \).

Would you like more details or clarifications on this?  

Here are some related questions to consider:

1. What are Pythagorean triplets, and how are they derived?
2. Can all integer values generate a Pythagorean triplet?
3. How does the formula for Pythagorean triplets work in generating primitive triplets?
4. Are there any other triplets where 8 is not the smallest value but still part of the set?
5. What are some real-life applications of Pythagorean triplets?

**Tip:** Primitive Pythagorean triplets are those where \( a, b, c \) have no common divisors other than 1.

Find a Pythagorean triplet whose smallest member is 8.

Discover how to find a Pythagorean triplet where the smallest member is 8 using the standard formula for generating triplets. This step-by-step solution covers key concepts in algebra and number theory, perfect for students in Grades 9-12.

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