To find the measure of side $XY$ in $\triangle XYZ$, given that $\triangle PQR \cong \triangle XYZ$ (congruent triangles), we need to find the corresponding side in $\triangle PQR$ to side $XY$. Since corresponding sides of congruent triangles have equal lengths, we first calculate the lengths of the sides of $\triangle PQR$.

Step 1: Find the length of $PQ$:

Points $P(0, 0)$ and $Q(-4, 0)$ lie on the x-axis. The distance between these points is the horizontal distance between them: $PQ = |x_2 - x_1| = |-4 - 0| = 4 \text{ units}$

Step 2: Find the length of $PR$:

Points $P(0, 0)$ and $R(0, 2)$ lie on the y-axis. The distance between these points is the vertical distance between them: $PR = |y_2 - y_1| = |2 - 0| = 2 \text{ units}$

Step 3: Find the length of $QR$:

Points $Q(-4, 0)$ and $R(0, 2)$ form a right triangle with the x-axis and y-axis. Using the distance formula: $QR = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(0 - (-4))^2 + (2 - 0)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47 \text{ units}$

Since $XY$ corresponds to $PQ$ in congruent triangles, and $PQ = 4$ units, the length of $XY$ is: $XY = 4 \text{ units}$

Thus, the measure of $XY$ is 4 units.

Would you like further details or clarifications?

Here are five related questions:

1. How do you determine which sides of two triangles correspond in congruent triangles?
2. How would the problem change if the triangles were similar but not congruent?
3. What is the perimeter of $\triangle PQR$?
4. Can the distance formula be used for three-dimensional coordinates?
5. How can the Pythagorean theorem be applied in this problem?

Tip: When working with congruent triangles, always match corresponding sides and angles based on the given vertices' order.