To calculate the length of the median $DC$, we can follow these steps:

Step 1: Find the midpoint $D$ of side $AB$.

The coordinates of the midpoint $D(x_1, y_1)$ of a line segment joining points $A(x_A, y_A)$ and $B(x_B, y_B)$ are given by:

$D = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right)$

Substituting the coordinates of $A(-3, 4)$ and $B(5, -2)$:

$D = \left(\frac{-3 + 5}{2}, \frac{4 + (-2)}{2}\right) = \left(\frac{2}{2}, \frac{2}{2}\right) = (1, 1)$

So, the coordinates of point $D$ are $(1, 1)$.

Step 2: Calculate the length of the median $DC$.

The length of a line segment $DC$ connecting points $D(x_1, y_1)$ and $C(x_C, y_C)$ is given by:

$DC = \sqrt{(x_C - x_1)^2 + (y_C - y_1)^2}$

Substituting the coordinates $D(1, 1)$ and $C(4, 7)$:

$DC = \sqrt{(4 - 1)^2 + (7 - 1)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$

Final Answer:

The length of the median $DC$ is $3\sqrt{5}$.

Do you want details on any step or have further questions?

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Related Questions:

1. How would you calculate the length of the other medians in this triangle?
2. Can you find the centroid of the triangle $ABC$?
3. What is the perimeter of triangle $ABC$?
4. How would you determine if triangle $ABC$ is right-angled?
5. What is the area of triangle $ABC$ using the coordinates of the vertices?

Tip: The centroid of a triangle divides each median into two segments, where the longer segment is twice the length of the shorter one.