Let's go through the solution step-by-step:

Given points:

• $P(1, 3)$
• $Q(1, 10)$

Part (a) Distance between $P$ and $Q$

The formula to find the distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is:

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

Using the given coordinates of $P$ and $Q$:

$d = \sqrt{(1 - 1)^2 + (10 - 3)^2}$

Simplifying:

$d = \sqrt{0 + (7)^2} = \sqrt{49} = 7$

So, the distance between $P$ and $Q$ is $7$ units.

Part (b) Midpoint of the line segment $PQ$

The formula for the midpoint of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is:

$\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Substitute the given coordinates of $P$ and $Q$:

$\text{Midpoint} = \left( \frac{1 + 1}{2}, \frac{3 + 10}{2} \right) = \left( \frac{2}{2}, \frac{13}{2} \right)$

Simplifying:

$\text{Midpoint} = (1, 6.5)$

So, the coordinates of the midpoint are $(1, 6.5)$.

Summary:

• (a) The distance between $P$ and $Q$ is $7$ units.
• (b) The midpoint of the line segment $PQ$ is $(1, 6.5)$.

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

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Here are 5 related questions:

1. How do you calculate the distance between two points in 3D space?
2. What is the formula for the midpoint if you are given points in 3D coordinates?
3. How would the distance change if the x-coordinates of $P$ and $Q$ were different?
4. Can the distance between two points ever be negative? Why or why not?
5. What is the significance of the midpoint in coordinate geometry?

Tip: Always check if the x or y coordinates of two points are the same. It simplifies the distance calculation.