To find the length of line segment $EG$, we can use the distance formula between two points in a coordinate plane. The formula is:

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

From the graph:

• The coordinates of point $E$ are $(-5, 6)$.
• The coordinates of point $G$ are $(3, -6)$.

Now, we can apply the distance formula:

$d = \sqrt{(3 - (-5))^2 + (-6 - 6)^2} = \sqrt{(3 + 5)^2 + (-6 - 6)^2} = \sqrt{8^2 + (-12)^2} = \sqrt{64 + 144} = \sqrt{208}$

$d \approx \sqrt{208} \approx 14.4 \text{ units}$

So, the length of $EG$ is approximately 14.4 units.

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

Here are 5 related questions to explore:

1. How do you calculate the length of a diagonal in a rectangle using coordinates?
2. What is the relationship between distance and slope of a line?
3. How can we generalize the distance formula to three dimensions?
4. How do you calculate the midpoint between two points?
5. What are the applications of the distance formula in real-world problems?

Tip: Always double-check coordinates when applying the distance formula to avoid simple mistakes.