To solve the problem, we need to find the length of each segment using the distance formula. The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Let's calculate the lengths of the segments $PQ$, $RS$, and $TV$.

1. For $PQ$, with endpoints $P(-6, 3)$ and $Q(-4, 0)$:

$PQ = \sqrt{((-4) - (-6))^2 + (0 - 3)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}$

2. For $RS$, with endpoints $R(-3, -4)$ and $S(0, -2)$:

$RS = \sqrt{(0 - (-3))^2 + (-2 - (-4))^2} = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}$

3. For $TV$, with endpoints $T(4, 2)$ and $V(5, -2)$:

$TV = \sqrt{(5 - 4)^2 + (-2 - 2)^2} = \sqrt{(1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}$

Answers:

• $PQ = \sqrt{13}$
• $RS = \sqrt{13}$
• $TV = \sqrt{17}$

Would you like any further details or have any questions?

Here are five related questions:

1. How do you apply the distance formula to find the length of a diagonal line in the coordinate plane?
2. What is the significance of the distance formula in determining the straight-line distance between two points?
3. How can the distance formula be used to prove that two segments are congruent?
4. How would the calculations change if the segments were in three-dimensional space?
5. What is the relationship between the distance formula and the Pythagorean Theorem?

Tip: To avoid mistakes in calculations, always ensure you square the differences correctly before summing them.