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 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 midpoints of the diagonals
The diagonals of a parallelogram bisect each other. We can find the midpoints of the diagonals formed by the pairs of opposite vertices.

- The midpoint \( M \) of the diagonal joining points \( (10, 45) \) and \( (28, 84) \):
\[ M = \left( \frac{10 + 28}{2}, \frac{45 + 84}{2} \right) = \left( \frac{38}{2}, \frac{129}{2} \right) = (19, 64.5) \]

- The midpoint \( N \) of the diagonal joining points \( (10, 14) \) and \( (28, 153) \):
\[ N = \left( \frac{10 + 28}{2}, \frac{14 + 153}{2} \right) = \left( \frac{38}{2}, \frac{167}{2} \right) = (19, 83.5) \]

### Step 2: Find the slope of the line through the origin and the midpoint
The line that divides the parallelogram into two congruent pieces will pass through the origin \( (0, 0) \) and the midpoint of the diagonals.

To find the slope \( m \) of the line through the origin and midpoint \( M \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{64.5 - 0}{19 - 0} = \frac{64.5}{19} \]

### Step 3: Simplify the slope
Now we simplify the slope:
\[ m = \frac{64.5}{19} = \frac{645}{190} = \frac{129}{38} \]

### Conclusion
The slope of the line that divides the parallelogram into two congruent pieces is:
\[ \boxed{\frac{129}{38}} \]