Question 4. Triangle ABC has vertices A(10, 2), B(4, −2), and C(−4, 6). Determine the equation of the median from the vertex A.

To determine the equation of the median from vertex \( A \) of triangle \( ABC \), follow these steps:

### Step 1: Find the midpoint of side \( BC \).
The median from vertex \( A \) will pass through the midpoint of the opposite side \( BC \). The coordinates of the midpoint \( M \) of a line segment with endpoints \( B(x_1, y_1) \) and \( C(x_2, y_2) \) can be found using the midpoint formula:

\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

For points \( B(4, -2) \) and \( C(-4, 6) \), the coordinates of the midpoint \( M \) are:

\[
M = \left(\frac{4 + (-4)}{2}, \frac{-2 + 6}{2}\right) = \left(\frac{0}{2}, \frac{4}{2}\right) = (0, 2)
\]

So, the midpoint \( M \) of side \( BC \) is \( (0, 2) \).

### Step 2: Determine the equation of the line passing through \( A(10, 2) \) and \( M(0, 2) \).
The equation of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be written in the point-slope form:

\[
y - y_1 = m(x - x_1)
\]

where \( m \) is the slope of the line, given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the points \( A(10, 2) \) and \( M(0, 2) \):

\[
m = \frac{2 - 2}{0 - 10} = \frac{0}{-10} = 0
\]

Since the slope \( m = 0 \), the line is horizontal. The equation of a horizontal line through the point \( (10, 2) \) (or any point on the line) is simply:

\[
y = 2
\]

### Final Answer:
The equation of the median from vertex \( A \) of triangle \( ABC \) is:

\[
y = 2
\]

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you find the equation of a median when the vertices have different y-coordinates?
2. What is the significance of the median in a triangle?
3. How can you find the centroid of a triangle using the medians?
4. How would the equation of the median change if point \( A \) had different coordinates?
5. How do you find the equation of a median in 3D space?

**Tip:** When working with triangles, always double-check your midpoint calculations, as this can affect the equation of the median.

Learn how to determine the equation of the median from vertex A in triangle ABC with a step-by-step solution. This problem involves finding the midpoint of a line segment and using the point-slope form of a line equation. Suitable for students in Grades 9-12.

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