`A(-2,4),B(-1,2),C(1,2)` and `D(2,4)` are the vertices of a quadrilateral. The line through vertex B which divides the quadrilateral into two equal areas has the equation

Options

A. `x+y=1`
B. `x-y+3=0`
C. `x+1=0`
D. none

Answer

To find the equation of the line through vertex B that divides the quadrilateral ABCD into two equal areas, we can follow these steps:

### Step 1: Identify the vertices of the quadrilateral
The vertices of the quadrilateral are given as:
- A(-2, 4)
- B(-1, 2)
- C(1, 2)
- D(2, 4)

### Step 2: Calculate the area of the quadrilateral ABCD
We can use the shoelace formula to find the area of the quadrilateral. The formula for the area \( A \) of a polygon given its vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is:

\[ A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| \]

where \((x_{n+1}, y_{n+1})\) is the same as \((x_1, y_1)\).

For our quadrilateral ABCD, we can arrange the vertices in order:
- A(-2, 4)
- B(-1, 2)
- C(1, 2)
- D(2, 4)
- A(-2, 4) (to close the polygon)

Now we can calculate the area:

\[ A = \frac{1}{2} \left| (-2 \cdot 2 + -1 \cdot 2 + 1 \cdot 4 + 2 \cdot 4) - (4 \cdot -1 + 2 \cdot 1 + 2 \cdot 2 + 4 \cdot -2) \right| \]

Calculating the terms:

\[ = \frac{1}{2} \left| (-4 - 2 + 4 + 8) - (-4 + 2 + 4 - 8) \right| \]
\[ = \frac{1}{2} \left| 6 - (-6) \right| \]
\[ = \frac{1}{2} \left| 6 + 6 \right| = \frac{1}{2} \cdot 12 = 6 \]

### Step 3: Find the area of each half
Since we need to divide the quadrilateral into two equal areas, each area will be:

\[ \text{Area of each half} = \frac{6}{2} = 3 \]

### Step 4: Determine the equation of the line through B
The line through point B(-1, 2) that divides the quadrilateral into two equal areas can be represented in the slope-intercept form \( y = mx + c \).

### Step 5: Find the slope of the line
To find the slope \( m \), we can consider the line that passes through point B and intersects the opposite side of the quadrilateral. We can check various slopes to find the one that divides the area into two equal parts.

### Step 6: Use the point-slope form to find the equation
Using the point-slope form of the line equation:

\[ y - y_1 = m(x - x_1) \]

Substituting \( B(-1, 2) \):

\[ y - 2 = m(x + 1) \]

### Step 7: Test various slopes
To find the correct slope \( m \), we can test various values and calculate the areas formed by the line until we find the one that gives us equal areas of 3.

After testing, we find that the line with the equation \( x - y = -3 \) (or equivalently \( x - y + 3 = 0 \)) divides the quadrilateral into two equal areas.

### Final Answer
The equation of the line through vertex B that divides the quadrilateral into two equal areas is:

\[ x - y = -3 \]