To determine the length of the segment $PQ$, we use the distance formula:

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

Here:

• Point $P$ has coordinates $(6, 12)$
• Point $Q$ has coordinates $(10, 5)$

Substitute these values into the formula:

$d = \sqrt{(10 - 6)^2 + (5 - 12)^2}$

Simplify:

$d = \sqrt{(4)^2 + (-7)^2}$ $d = \sqrt{16 + 49}$ $d = \sqrt{65}$

The length of segment $PQ$ is approximately:

$\sqrt{65} \approx 8.06 \text{ units}$

Let me know if you'd like more details or clarifications.

Related Questions:

1. How can we find the distance between any two points in 3D space?
2. What is the geometric interpretation of the distance formula?
3. Can you explain how the Pythagorean Theorem is applied in this formula?
4. How would the length of $PQ$ change if $P$ or $Q$ were moved?
5. How can this be used to find the perimeter of a polygon?

Tip:

Always double-check coordinates before substituting into formulas to avoid calculation errors!