To find the distance between points $P(4, -7)$ and $Q(5, -3)$, we use the distance formula:

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

For points $P(4, -7)$ and $Q(5, -3)$:

• $x_1 = 4$, $y_1 = -7$
• $x_2 = 5$, $y_2 = -3$

Substitute these values into the formula:

$d = \sqrt{(5 - 4)^2 + (-3 + 7)^2}$

Calculate each part:

1. $(5 - 4)^2 = 1^2 = 1$
2. $(-3 + 7)^2 = 4^2 = 16$

Now, substitute back:

$d = \sqrt{1 + 16} = \sqrt{17}$

Thus, the correct answer is A. $\sqrt{17}$.

Would you like further details, or do you have any questions?

Here are some related questions to deepen your understanding:

1. What is the general form of the distance formula for two points in 3D space?
2. How would the distance formula change if we were using points in polar coordinates?
3. Can you explain why distance is always a positive value?
4. How can you apply the distance formula to find the length of the hypotenuse in a right triangle?
5. How can we find the midpoint between two points $P$ and $Q$?

Tip: Always ensure your values are correctly substituted into formulas to avoid simple arithmetic errors!