We are asked to show that the midpoints of the sides of a rectangle form a rhombus. We are asked to write a coordinate proof, so let's place the rectangle in the corner of the first quadrant of the coordinate plane so that it's sides are parallel to the coordinate axes.
Since the sides are parallel to the coordinate axes, if the coordinates of B are (0,b) and the coordinates of D are (d,0), then the coordinates of C are (d,b). We are asked to investigate the midpoints of the sides, so let's recall the Midpoint Formula.
The midpoint between points (x_1,y_1) and (x_2,y_2) is ( x_1+x_22, y_1+y_22).
Let's substitute the coordinates of the vertices to this formula to find the midpoints of the sides.
Endpoints
Substitution
Midpoint
A(0,0) and B(0,b)
(0+0/2,0+b/2)
M_(AB)(0,b/2)
B(0,b) and C(d,b)
(0+d/2,b+b/2)
M_(BC)(d/2,b)
C(d,b) and D(d,0)
(d+d/2,b+0/2)
M_(CD)(d,b/2)
D(d,0) and A(0,0)
(d+0/2,0+0/2)
M_(DA)(d/2,0)
We can use the Distance Formula to find the length of the sides of the quadrilateral formed by these midpoints.
The distance between points (x_1,y_1) and (x_2,y_2) is sqrt((x_2-x_1)^2+(y_2-y_1)^2).
Let's start with the distance between midpoints M_(AB) and M_(BC).
We see that the sides of quadrilateral M_(AB)M_(BC)M_(CD)M_(DA) have equal length, so they are congruent. Since opposite sides are congruent, Theorem 6.9 says that the quadrilateral is a parallelogram. Since M_(AB)M_(BC)M_(CD)M_(DA) is a parallelogram with congruent sides, by definition, it is a rhombus.
