To prove that all three medians of triangle A(4, 5), B(1, 2), and C(6, 2) meet at the same point, the medians of each side must be found. What are the three median bisecotrs.

Answer 1

To find the medians of triangle \( ABC \) with vertices \( A(4, 5) \), \( B(1, 2) \), and \( C(6, 2) \), we first need to find the midpoints of each side. A median connects each vertex of the triangle to the midpoint of the opposite side.

Step 1: Find the midpoints of the sides

1. Midpoint of side \( BC \): \[ M_{BC} = \left( \frac{B_x + C_x}{2}, \frac{B_y + C_y}{2} \right) = \left( \frac{1 + 6}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = (3.5, 2) \]

2. Midpoint of side \( AC \): \[ M_{AC} = \left( \frac{A_x + C_x}{2}, \frac{A_y + C_y}{2} \right) = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( 5, \frac{7}{2} \right) = (5, 3.5) \]

3. Midpoint of side \( AB \): \[ M_{AB} = \left( \frac{A_x + B_x}{2}, \frac{A_y + B_y}{2} \right) = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = (2.5, 3.5) \]

Step 2: Find the equations of the medians

Next, we calculate the equations of the medians, which are the lines connecting each vertex to the midpoint of the opposite side.

1. Median from \( A \) to \( M_{BC} \):

   • Points: \( A(4, 5) \) and \( M_{BC}(3.5, 2) \)
   • Slope: \[ m_{AM_{BC}} = \frac{2 - 5}{3.5 - 4} = \frac{-3}{-0.5} = 6 \]
   • Equation (using point-slope form with point \( A \)): \[ y - 5 = 6(x - 4) \implies y = 6x - 24 + 5 \implies y = 6x - 19 \]

2. Median from \( B \) to \( M_{AC} \):

   • Points: \( B(1, 2) \) and \( M_{AC}(5, 3.5) \)
   • Slope: \[ m_{BM_{AC}} = \frac{3.5 - 2}{5 - 1} = \frac{1.5}{4} = \frac{3}{8} \]
   • Equation (using point-slope form with point \( B \)): \[ y - 2 = \frac{3}{8}(x - 1) \implies 8y - 16 = 3x - 3 \implies 3x - 8y + 13 = 0 \]

3. Median from \( C \) to \( M_{AB} \):

   • Points: \( C(6, 2) \) and \( M_{AB}(2.5, 3.5) \)
   • Slope: \[ m_{CM_{AB}} = \frac{3.5 - 2}{2.5 - 6} = \frac{1.5}{-3.5} = -\frac{3}{7} \]
   • Equation (using point-slope form with point \( C \)): \[ y - 2 = -\frac{3}{7}(x - 6) \implies 7y - 14 = -3x + 18 \implies 3x + 7y - 32 = 0 \]

Step 3: Find the intersection of the medians

The three equations of the medians are:

1. \( y = 6x - 19 \) (Median from \( A \))
2. \( 3x - 8y + 13 = 0 \) (Median from \( B \))
3. \( 3x + 7y - 32 = 0 \) (Median from \( C \))

These medians intersect at the centroid of the triangle, which can be found by the formula: \[ G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) = \left( \frac{4 + 1 + 6}{3}, \frac{5 + 2 + 2}{3} \right) = \left( \frac{11}{3}, \frac{9}{3} \right) = \left( \frac{11}{3}, 3 \right) \]

Final Answer

All three medians intersect at the same point, the centroid, which is \( \left( \frac{11}{3}, 3 \right) \).