Question

Matikas is writing a coordinate proof to show that the midpoints of a quadrilateral are the vertices of a parallelogram. He starts by assigning coordinates to the vertices of quadrilateral RSTV
and labeling the midpoints of the sides of the quadrilateral as A, B, C, and D.

Quadrilateral R S T V in a coordinate plane with vertex R at 0 comma 0, vertex S in the first quadrant at 2 a comma 2b, vertex T also in the first quadrant at 2 c comma 2 d, and vertex V on the positive side of the x-axis at 2 c comma 0. Point A is between points R and S, point B is between points S and T, point C is between points T and V, and point D is between points R and V.

Enter the answers, in simplified form, by filling in the boxes to complete the proof.

The coordinates of point A are (
,
).

The coordinates of point B are (a+c, b+d)
.

The coordinates of point C are (
,
).

The coordinates of point D are (c, 0)
.

The slope of both AB¯¯¯¯¯
and DC¯¯¯¯¯
is dc
.

The slope of both AD¯¯¯¯¯
and BC¯¯¯¯¯
is
.

Because both pairs of opposite sides are parallel, quadrilateral ABCD
is a parallelogram.

Answer 1

The coordinates of point A are (a, b).

The coordinates of point B are (a + c, b + d).

The coordinates of point C are (2c, d).

The coordinates of point D are (c, 0).

The slope of both AB and DC is d/c.

The slope of both AD and BC is -b/(c - a).

Because both pairs of opposite sides are parallel, quadrilateral ABCD is a parallelogram.