Question

The coordinates LMN L(4,7) M(6,-1) N(-2,3) determine equations of two medians and algebraic ally solve the system of equations represented by the medians to determine the location of the centroid

Answer 1

To determine the centroid of triangle \(LMN\) formed by points \(L(4,7)\), \(M(6,-1)\), and \(N(-2,3)\), we can first find the midpoints of two sides of the triangle and then determine the equations of the medians from the vertices to those midpoints.

### Step 1: Find Midpoints

1. **Midpoint of side \(LM\)**:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{4 + 6}{2}, \frac{7 - 1}{2} \right) = \left( \frac{10}{2}, \frac{6}{2} \right) = (5, 3)
\]

2. **Midpoint of side \(LN\)**:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{4 - 2}{2}, \frac{7 + 3}{2} \right) = \left( \frac{2}{2}, \frac{10}{2} \right) = (1, 5)
\]

### Step 2: Write the equations of the medians

1. **Median from \(M\) to the midpoint of \(LN\) (1, 5)**:
- Points \(M(6,-1)\) and midpoint \((1,5)\).
- The slope \(m\) from \(M\) to the midpoint is calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{1 - 6} = \frac{6}{-5} = -\frac{6}{5}
\]
- Using point-slope form of a line (\(y - y_1 = m(x - x_1)\)):
\[
y - (-1) = -\frac{6}{5}(x - 6)
\]
\[
y + 1 = -\frac{6}{5}x + \frac{36}{5}
\]
\[
y = -\frac{6}{5}x + \frac{36}{5} - 1 \quad \text{(convert \(-1\) to fraction)}
\]
\[
y = -\frac{6}{5}x + \frac{36}{5} - \frac{5}{5}
\]
\[
y = -\frac{6}{5}x + \frac{31}{5}
\]

2. **Median from \(N\) to the midpoint of \(LM\) (5, 3)**:
- Points \(N(-2,3)\) and midpoint \((5, 3)\).
- The slope \(m\) from \(N\) to the midpoint:
\[
m = \frac{3 - 3}{5 - (-2)} = \frac{0}{7} = 0
\]
- The equation of the median is:
\[
y = 3 \quad \text{(horizontal line passing through } (5, 3)\text{)}
\]

### Step 3: Solve for the intersecting point of medians

Set the equations equal:
1. \(y = -\frac{6}{5}x + \frac{31}{5}\)
2. \(y = 3\)

Substituting \(y = 3\) into the median equation:
\[
3 = -\frac{6}{5}x + \frac{31}{5}
\]
Multiply through by 5 to eliminate the fraction:
\[
15 = -6x + 31
\]
Rearranging gives:
\[
6x = 31 - 15
\]
\[
6x = 16
\]
\[
x = \frac{16}{6} = \frac{8}{3}
\]

Substituting \(x = \frac{8}{3}\) back into \(y = 3\):
\[
y = 3
\]

### Final Result: Centroid

The centroid \(G\) of triangle \(LMN\) is at the coordinates:
\[
G\left(\frac{8}{3}, 3\right)
\]