In a `DeltaABC, A = (2,3)` and medians through B and C have equations `x +y - 1 = 0` and `2y - 1 = 0`
Equation of median through A is

Options

A. `x +y = 4`
B. `5x - 3y = 1`
C. `5x +3y = 1`
D. `5x = 3y`

Explanation

To find the equation of the median through point A in triangle ABC, we follow these steps:

### Step 1: Identify the given points and equations
We are given:
- Point A = (2, 3)
- Equation of the median through B: \(x + y - 1 = 0\)
- Equation of the median through C: \(2y - 1 = 0\)

### Step 2: Find the coordinates of point F (centroid)
The centroid F of triangle ABC can be found by solving the equations of the medians through points B and C.

1. From the equation \(2y - 1 = 0\):
\[ 2y = 1 \implies y = \frac{1}{2} \]

2. Substitute \(y = \frac{1}{2}\) into the first equation \(x + y - 1 = 0\):
\[ x + \frac{1}{2} - 1 = 0 \implies x = 1 - \frac{1}{2} = \frac{1}{2} \]

Thus, the coordinates of point F (centroid) are:
\[ F = \left(\frac{1}{2}, \frac{1}{2}\right) \]

### Step 3: Calculate the slope of the line AF
To find the slope of the line passing through points A and F, we use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of A (2, 3) and F \(\left(\frac{1}{2}, \frac{1}{2}\right)\):
\[ m = \frac{\frac{1}{2} - 3}{\frac{1}{2} - 2} = \frac{\frac{1}{2} - \frac{6}{2}}{\frac{1}{2} - \frac{4}{2}} = \frac{-\frac{5}{2}}{-\frac{3}{2}} = \frac{5}{3} \]

### Step 4: Use point-slope form to find the equation of line AF
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = \frac{5}{3}\), \(x_1 = 2\), and \(y_1 = 3\):
\[ y - 3 = \frac{5}{3}(x - 2) \]
Expanding this:
\[ y - 3 = \frac{5}{3}x - \frac{10}{3} \]
Rearranging gives:
\[ y = \frac{5}{3}x - \frac{10}{3} + 3 \]
Converting 3 into a fraction:
\[ 3 = \frac{9}{3} \]
Thus:
\[ y = \frac{5}{3}x - \frac{10}{3} + \frac{9}{3} = \frac{5}{3}x - \frac{1}{3} \]

### Step 5: Write the equation in standard form
To convert to standard form \(Ax + By + C = 0\):
\[ \frac{5}{3}x - y - \frac{1}{3} = 0 \]
Multiplying through by 3 to eliminate the fractions:
\[ 5x - 3y - 1 = 0 \]
Rearranging gives:
\[ 5x - 3y = 1 \]

### Final Answer
The equation of the median through point A is:
\[ 5x - 3y = 1 \]

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