Question

Vertices of a parallelogram ABCD are A(3, 1), B(13, 6), C(13, 21) and D(3, 16). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is

• A. $\frac{11}{12}$
• B. $\frac{11}{8}$
• C. $\frac{25}{8}$
• D. $\frac{13}{8}$

Answer 1

Answer: (B)

$\frac{11}{8}$

Answer 2

A line that divides a parallelogram into two congruent parts must pass through the center of the parallelogram. The center of a parallelogram is the point of intersection of its diagonals, which is also the midpoint of each diagonal.

The coordinates of the vertices are A(3, 1), B(13, 6), C(13, 21), and D(3, 16). We calculate the midpoint of the diagonal AC using the midpoint formula $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. Center $M = \left(\frac{3+13}{2}, \frac{1+21}{2}\right) = \left(\frac{16}{2}, \frac{22}{2}\right) = (8, 11)$.

The line passes through the origin (0, 0) and the center of the parallelogram (8, 11). The slope of this line is calculated using the slope formula $m = \frac{y_2-y_1}{x_2-x_1}$. Slope $m = \frac{11-0}{8-0} = \frac{11}{8}$.