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!