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We are asked to find the distance between points \( A(2, 1) \) and \( B(-4, 7) \). The formula for the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the coordinate plane is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Substitute the coordinates into the formula.
- \( x_1 = 2 \), \( y_1 = 1 \)
- \( x_2 = -4 \), \( y_2 = 7 \)

\[
d = \sqrt{((-4) - 2)^2 + (7 - 1)^2}
\]
\[
d = \sqrt{(-6)^2 + (6)^2}
\]
\[
d = \sqrt{36 + 36}
\]
\[
d = \sqrt{72}
\]
\[
d \approx 8.5 \quad \text{(rounded to the nearest tenth)}
\]

Thus, the distance \( d \) is approximately **8.5**.

Would you like a detailed explanation or have any questions?

Here are 5 related questions to further explore this concept:
1. How would you find the midpoint of segment \( AB \)?
2. How can you use the distance formula in three-dimensional space?
3. What is the equation of a circle with center \( A(2, 1) \) and radius equal to the distance \( d \)?
4. How does the Pythagorean theorem relate to the distance formula?
5. How do you find the slope of the line passing through points \( A \) and \( B \)?

**Tip:** The distance formula is essentially the Pythagorean theorem applied in the coordinate plane!

Find the distance, d, between points A(2, 1) and B(-4, 7).

Learn how to find the distance between points A(2, 1) and B(-4, 7) using the distance formula. This detailed step-by-step solution shows the application of the Pythagorean theorem and the coordinate plane. Ideal for students in Grades 8-10.

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