A line through the origin divides parallelogram with vertices (10,45) , (10, 14) , (28,153) and (28,84) into two congruent pieces. The slope of the line is

Options

A. `19/99`
B. `167/38`
C. `9/19`
D. `19/9`

Answer

To find the slope of the line that passes through the origin and divides the parallelogram with vertices (10, 45), (10, 14), (28, 153), and (28, 84) into two congruent pieces, we can follow these steps:

### Step 1: Identify the vertices of the parallelogram
The vertices of the parallelogram are given as:
- A(10, 45)
- B(10, 14)
- C(28, 153)
- D(28, 84)

### Step 2: Determine the midpoints of the diagonals
To find the slope of the line that divides the parallelogram into two congruent pieces, we first need to find the midpoints of the diagonals AC and BD.

**Midpoint of AC:**
\[ M_{AC} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{10 + 28}{2}, \frac{45 + 153}{2}\right) = \left(19, 99\right) \]

**Midpoint of BD:**
\[ M_{BD} = \left(\frac{10 + 28}{2}, \frac{14 + 84}{2}\right) = \left(19, 49\right) \]

### Step 3: Find the slope of the line through the origin and the midpoints
The slope \( m \) of the line through the origin (0, 0) and the midpoint \( M_{AC} \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For the midpoint \( M_{AC} \):
\[ m_{AC} = \frac{99 - 0}{19 - 0} = \frac{99}{19} \]

For the midpoint \( M_{BD} \):
\[ m_{BD} = \frac{49 - 0}{19 - 0} = \frac{49}{19} \]

### Step 4: Calculate the slope of the line that divides the parallelogram
Since the line divides the parallelogram into two congruent pieces, we need to find the average of the slopes of the two midpoints:
\[ m = \frac{m_{AC} + m_{BD}}{2} = \frac{\frac{99}{19} + \frac{49}{19}}{2} = \frac{\frac{148}{19}}{2} = \frac{148}{38} = \frac{74}{19} \]

### Conclusion
The slope of the line that divides the parallelogram into two congruent pieces is:
\[ \text{slope} = \frac{74}{19} \]