Find the equation of the medians of `DeltaABC` whose vertices are `A(1,0)`, `B(2,4)` and `C(3,2)`.

Answer

To find the equations of the medians of triangle ABC with vertices A(1,0), B(2,4), and C(3,2), we will follow these steps:

### Step 1: Find the midpoints of the sides of the triangle

1. **Midpoint of BC (let's call it P)**:
- Coordinates of B = (2, 4)
- Coordinates of C = (3, 2)
- Midpoint formula: \( P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- \( P = \left( \frac{2 + 3}{2}, \frac{4 + 2}{2} \right) = \left( \frac{5}{2}, 3 \right) \)

2. **Midpoint of AC (let's call it Q)**:
- Coordinates of A = (1, 0)
- Coordinates of C = (3, 2)
- \( Q = \left( \frac{1 + 3}{2}, \frac{0 + 2}{2} \right) = \left( 2, 1 \right) \)

3. **Midpoint of AB (let's call it R)**:
- Coordinates of A = (1, 0)
- Coordinates of B = (2, 4)
- \( R = \left( \frac{1 + 2}{2}, \frac{0 + 4}{2} \right) = \left( \frac{3}{2}, 2 \right) \)

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

1. **Equation of median AP**:
- Points A(1, 0) and P(5/2, 3)
- Using the formula \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \):
- \( y - 0 = \frac{3 - 0}{\frac{5}{2} - 1}(x - 1) \)
- Simplifying: \( y = \frac{3}{\frac{3}{2}}(x - 1) = 2(x - 1) \)
- Rearranging gives: \( 2x - y - 2 = 0 \)

2. **Equation of median BQ**:
- Points B(2, 4) and Q(2, 1)
- Since both points have the same x-coordinate, the equation is simply \( x = 2 \).

3. **Equation of median CR**:
- Points C(3, 2) and R(3/2, 2)
- Since both points have the same y-coordinate, the equation is simply \( y = 2 \).

### Final Equations of the Medians
- Median AP: \( 2x - y - 2 = 0 \)
- Median BQ: \( x = 2 \)
- Median CR: \( y = 2 \)

---