Lucinda is writing a coordinate proof to show that a diagonal of a parallelogram partitions the parallelogram into two equal areas.
A parallelogram graphed on a coordinate plane. The vertices of rectangle are labeled as K L M and N. The vertex labeled as K lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as L lies on begin ordered pair x comma 2 y end ordered pair. The coordinate of vertex M is left blank. The vertex labeled as N lies on begin ordered pair 3 x comma 0 end ordered pair. A diagonal is drawn between points K and M.
Enter your answers in the boxes to complete Lucinda's proof.
Since KLMN is a parallelogram and a parallelogram's opposite sides are parallel and congruent, the coordinates for M are (4x, 2y).
In △KMN, the length of the base is and the height is . So an expression for the area of △KMN is .
In △KLM, the length of the base is 3x and the height is 2y. So an expression for the area of △KLM is .
Comparing the area of the two triangles that are formed by a diagonal of the parallelogram shows that a diagonal of a parallelogram partitions the parallelogram into two equal areas.
