To solve the problem step by step, let's first summarize the information provided:
1. Triangle ABC has sides AB parallel to the y-axis and BC parallel to the x-axis.
2. The centroid G of triangle ABC is at (2, 1).
3. The median through vertex C is given by the equation \(x - y = 1\).
4. We need to find the slope of the median through vertex A.
### Step 1: Identify the coordinates of the vertices
Since AB is parallel to the y-axis, we can assume:
- A = (a, b)
- B = (a, d) (since both A and B share the same x-coordinate)
- C = (c, b) (since BC is parallel to the x-axis, both B and C share the same y-coordinate)
### Step 2: Find the coordinates of the centroid
The coordinates of the centroid G of triangle ABC can be calculated using the formula:
\[
G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]
Substituting the coordinates of A, B, and C:
\[
G = \left(\frac{a + a + c}{3}, \frac{b + d + b}{3}\right) = \left(\frac{2a + c}{3}, \frac{2b + d}{3}\right)
\]
Given that the centroid G is at (2, 1), we can set up the equations:
\[
\frac{2a + c}{3} = 2 \quad \text{(1)}
\]
\[
\frac{2b + d}{3} = 1 \quad \text{(2)}
\]
### Step 3: Solve for c and d
From equation (1):
\[
2a + c = 6 \quad \Rightarrow \quad c = 6 - 2a \quad \text{(3)}
\]
From equation (2):
\[
2b + d = 3 \quad \Rightarrow \quad d = 3 - 2b \quad \text{(4)}
\]
### Step 4: Find the slope of the median through C
The median through C connects C to the midpoint D of AB. The coordinates of D can be calculated as:
\[
D = \left(a, \frac{b + d}{2}\right) = \left(a, \frac{b + (3 - 2b)}{2}\right) = \left(a, \frac{3 - b}{2}\right)
\]
The slope of the median through C (from C to D) is given by:
\[
\text{slope of CD} = \frac{\frac{3 - b}{2} - b}{a - c}
\]
Substituting \(c\) from equation (3):
\[
\text{slope of CD} = \frac{\frac{3 - b}{2} - b}{a - (6 - 2a)} = \frac{\frac{3 - b}{2} - b}{3a - 6}
\]
Given that the median through C is \(x - y = 1\), we can find its slope:
\[
\text{slope} = 1
\]
Thus, we have:
\[
\frac{\frac{3 - b}{2} - b}{3a - 6} = 1
\]
### Step 5: Solve for b and a
Cross-multiplying gives:
\[
\frac{3 - b}{2} - b = 3a - 6
\]
Simplifying:
\[
\frac{3 - 3b}{2} = 3a - 6
\]
Multiplying through by 2:
\[
3 - 3b = 6a - 12
\]
Rearranging gives:
\[
6a + 3b = 15 \quad \text{(5)}
\]
### Step 6: Find the slope of the median through A
The slope of the median through A (from A to the midpoint E of BC) is:
\[
E = \left(\frac{a + c}{2}, b\right) = \left(\frac{a + (6 - 2a)}{2}, b\right) = \left(\frac{6 - a}{2}, b\right)
\]
The slope of median through A is:
\[
\text{slope of AE} = \frac{b - b}{\frac{6 - a}{2} - a} = \frac{0}{\frac{6 - 3a}{2}} = 0
\]
### Final Answer
The slope of the median through A is \(0\).
