P(3, 4), Q(7, -2) and R(-2, -1) are the vertices of triangle PQR. Write down the equation of the median of the triangle through R. 

Answer

To find the equation of the median of triangle PQR through vertex R, we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of triangle PQR are:
- P(3, 4)
- Q(7, -2)
- R(-2, -1)

### Step 2: Find the midpoint D of side PQ
To find the midpoint D of line segment PQ, we use the midpoint formula:
\[ D\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Where \(P(x_1, y_1) = (3, 4)\) and \(Q(x_2, y_2) = (7, -2)\).

Calculating the coordinates of D:
\[ D\left(\frac{3 + 7}{2}, \frac{4 + (-2)}{2}\right) = D\left(\frac{10}{2}, \frac{2}{2}\right) = D(5, 1) \]

### Step 3: Use the two-point form of the line equation
Now, we need to find the equation of the line (median) that passes through points R and D. The formula for the equation of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Using R(-2, -1) as \((x_1, y_1)\) and D(5, 1) as \((x_2, y_2)\):
\[ y - (-1) = \frac{1 - (-1)}{5 - (-2)}(x - (-2)) \]

### Step 4: Simplify the equation
Calculating the slope:
\[ y + 1 = \frac{1 + 1}{5 + 2}(x + 2) = \frac{2}{7}(x + 2) \]

Expanding the equation:
\[ y + 1 = \frac{2}{7}x + \frac{4}{7} \]

### Step 5: Rearranging to standard form
To rearrange it into standard form \(Ax + By + C = 0\):
1. Multiply through by 7 to eliminate the fraction:
\[ 7(y + 1) = 2x + 4 \]
\[ 7y + 7 = 2x + 4 \]
2. Rearranging gives:
\[ 2x - 7y + 3 = 0 \]

### Final Equation
Thus, the equation of the median through point R is:
\[ 2x - 7y = -3 \]