The question in the image asks for the length of segment $AE$, given that segments $AB$ and $CD$ meet at point $E$. The coordinates of the points are as follows:

• $A(-3, 2)$
• $B(3, -1)$
• $C(1, 1)$
• $D(-1, -1)$
• $E(0, 0)$ (intersection point of segments $AB$ and $CD$)

To find the length of segment $AE$, we will calculate the distance between points $A(-3, 2)$ and $E(0, 0)$.

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

For segment $AE$, using points $A(-3, 2)$ and $E(0, 0)$:

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

Therefore, the length of segment $AE$ is $\sqrt{13}$ units.

Would you like more details or have any questions? Here are some related questions to explore:

1. How can we find the equation of line $AB$?
2. How can we find the equation of line $CD$?
3. How do we determine the coordinates of the intersection point $E$ algebraically?
4. What is the length of segment $AB$?
5. Can the lengths of all segments be expressed as exact square roots?

Tip: When working with lattice points on a coordinate plane, the distance formula is a crucial tool to calculate the length between any two points.