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: \[ G\left(\frac{8}{3}, 3\right) \]