https://file.aiquickdraw.com/mathbot/file/172621033854528z67t17.jpg

To solve the problem, we need to find the equation of the median of a triangle. Here's the process:

1. **Vertices of the triangle**: 
   - \( A(4, 1) \)
   - \( B(-12, 8) \)
   - \( C(-2, -4) \)

2. **Midpoint of side BC**: 
   The median is drawn from vertex \( A(4, 1) \) to the midpoint of side \( BC \). So, we first need to find the midpoint of line segment \( BC \).

   The formula for the midpoint of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
   \[
   \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Applying this to points \( B(-12, 8) \) and \( C(-2, -4) \):
   \[
   \text{Midpoint} = \left( \frac{-12 + (-2)}{2}, \frac{8 + (-4)}{2} \right) = \left( \frac{-14}{2}, \frac{4}{2} \right) = (-7, 2)
   \]

3. **Equation of the median**: 
   Now, the median is the line passing through points \( A(4, 1) \) and the midpoint \( (-7, 2) \). To find the equation of this line, we need to compute the slope between these two points.

   The slope \( m \) is given by the formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Using points \( A(4, 1) \) and \( (-7, 2) \):
   \[
   m = \frac{2 - 1}{-7 - 4} = \frac{1}{-11} = -\frac{1}{11}
   \]

4. **Equation of the line**: 
   Using the point-slope form of the line equation:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = -\frac{1}{11} \) and point \( A(4, 1) \):
   \[
   y - 1 = -\frac{1}{11}(x - 4)
   \]
   Expanding this equation:
   \[
   y - 1 = -\frac{1}{11}x + \frac{4}{11}
   \]
   \[
   y = -\frac{1}{11}x + \frac{4}{11} + 1
   \]
   \[
   y = -\frac{1}{11}x + \frac{4}{11} + \frac{11}{11} = -\frac{1}{11}x + \frac{15}{11}
   \]

Thus, the equation of the median is:
\[
y = -\frac{1}{11}x + \frac{15}{11}
\]
This equation is valid for the \( x \)-range between the x-coordinates of the points \( A(4, 1) \) and the midpoint \( (-7, 2) \), so \( -7 \leq x \leq 4 \).

### Final Answer:
\[
y = -\frac{1}{11}x + \frac{15}{11} \quad \text{for} \quad -7 \leq x \leq 4
\]

---

Would you like further details or have any questions?

Here are some related questions:
1. How do you find the equation of a line given two points?
2. What is the geometric significance of the median in a triangle?
3. How do you calculate the midpoint of a line segment?
4. What is the point-slope form of the equation of a line?
5. Can a triangle have more than one median, and how are they related?

**Tip**: To check if two lines are parallel, compare their slopes. If they are equal, the lines are parallel!

Find an equation of the median of a triangle drawn from the vertex A(4, 1) to the side formed by B(-12, 8) and C(-2, -4).

This problem involves finding the equation of the median of a triangle drawn from vertex A(4,1) to the side BC formed by points B(-12, 8) and C(-2, -4). It requires calculating the midpoint of BC, finding the slope, and using point-slope form to derive the equation of the median.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation