The vertices of a triangle are the points A (-3,4); B (5,-2) & C (4,7). If D is the midpoint of side AB, calculate the lenght of the median DC.

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.

Learn how to calculate the length of median DC in triangle ABC with vertices at points A (-3, 4), B (5, -2), and C (4, 7). Follow step-by-step instructions using the midpoint formula and distance formula. Suitable for Grades 9-12 students studying coordinate geometry.

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