The vertices of a parallelogram `A B C D` are `A(3,1),B(13 ,6),C(13 ,21),` and `D(3,16)dot` If a line passing through the origin divides the parallelogram into two congruent parts, then the slope of the line is (a) `(11)/(12)` (b) `(11)/8` (c) `(25)/8` (d) `(13)/8`

Options

A. `11//12`
B. `11//8`
C. `25//8`
D. `13//8`

Explanation

To solve the problem, we need to find the slope of a line passing through the origin that divides the parallelogram ABCD into two congruent parts. The vertices of the parallelogram are given as A(3, 1), B(13, 6), C(13, 21), and D(3, 16).

### Step-by-Step Solution:

1. **Identify the Midpoints of the Diagonals:**
The diagonals of a parallelogram bisect each other. We will find the midpoints of the diagonals AC and BD.

- **Midpoint of AC:**
\[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{3 + 13}{2}, \frac{1 + 21}{2} \right) = \left( \frac{16}{2}, \frac{22}{2} \right) = (8, 11) \]

- **Midpoint of BD:**
\[ M_{BD} = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) = \left( \frac{13 + 3}{2}, \frac{6 + 16}{2} \right) = \left( \frac{16}{2}, \frac{22}{2} \right) = (8, 11) \]

Since both midpoints are the same, we confirm that the diagonals bisect each other at point (8, 11).

2. **Find the Slope of the Line through the Origin to the Midpoint:**
The slope \( m \) of a line passing through the origin (0, 0) and the midpoint (8, 11) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 0}{8 - 0} = \frac{11}{8} \]

3. **Conclusion:**
The slope of the line that divides the parallelogram ABCD into two congruent parts is \( \frac{11}{8} \).

### Final Answer:
The slope of the line is \( \frac{11}{8} \), which corresponds to option (b).

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