Let's break down and solve the problem step by step.

Given Information:

• Coordinates of $X = -7$
• Coordinates of $Y = -3$
• Coordinates of $Z = 4$
• Coordinates of $W = 10$

Required:

• Find the lengths of line segments $\overline{XY}$ and $\overline{ZW}$.
• Determine if these segments are congruent (i.e., have equal lengths).

Step 1: Find the Length of $\overline{XY}$

The length of a line segment between two points on a number line is simply the absolute difference between their coordinates. So,

$\text{Length of } \overline{XY} = |X - Y| = |-7 - (-3)| = |-7 + 3| = |-4| = 4$

Step 2: Find the Length of $\overline{ZW}$

Similarly, for $\overline{ZW}$,

$\text{Length of } \overline{ZW} = |Z - W| = |4 - 10| = |-6| = 6$

Step 3: Compare the Two Segments

• The length of $\overline{XY}$ is 4.
• The length of $\overline{ZW}$ is 6.

Since $4 \neq 6$, the two segments $\overline{XY}$ and $\overline{ZW}$ are not congruent.

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

Here are 5 related questions you can explore:

1. How do you find the midpoint of a segment on a number line?
2. What is the formula for the distance between two points in a coordinate plane?
3. What are congruent segments, and what are some examples in geometry?
4. How would you find the length of a segment if it involved coordinates in 2D space?
5. Can two segments have different coordinates and still be congruent?

Tip: The absolute value ensures that the length of a segment is always positive, no matter the direction on the number line.