Exercise 19 Page 417

We are asked to prove 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 rectangular flag in a coordinate plane so that two of its sides are along the coordinate axes.

Before deciding on the expressions for the coordinates of the vertices of rectangle $ABCD,$ let's recall some properties.

• Side $\overline{BC}$ is parallel to the $x\text{-}$axis, so the $y\text{-}$coordinates of $B$ and $C$ are the same.
• Side $\overline{CD}$ is parallel to the $y\text{-}$axis, so the $x\text{-}$coordinates of $C$ and $D$ are the same.

Since we also need the coordinates of the midpoints of the sides, we will use $B(0,2b),$ $C(2d,2b),$ and $D(2d,0)$ for the vertices.

On vertical segments, the midpoint has the same $x\text{-}$coordinate as the two endpoints and the $y\text{-}$coordinate is the average.

• The midpoint between points $A(0,0)$ and $B(0,2b)$ is $M_{AB}(0,b).$
• The midpoint between points $C(2d,2b)$ and $D(2d,0)$ is $M_{CD}(2d,b).$

On horizontal segments, the midpoint has the same $y\text{-}$coordinate as the two endpoints and the $x\text{-}$coordinate is the average.

• The midpoint between points $D(2d,0)$ and $A(0,0)$ is $M_{DA}(d,0).$
• The midpoint between points $B(0,2b)$ and $C(2d,2b)$ is $M_{BC}(d,2b).$

We can again use our observations above to find the midpoints of the diagonals.

• The midpoint between points $M_{AB}(0,b)$ and $M_{CD}(2d,b)$ is $(d,b).$
• The midpoint between points $M_{BC}(d,2b)$ and $M_{DA}(d,0)$ is $(d,b).$

Let's collect what we know about diagonals $\overline{M_{AB}{M}_{CD}}$ and $\overline{M_{BC}{M}_{DA}}$

Let's finish the proof by recalling two theorems.

• Since the diagonals bisect each other, according to Theorem $6\text{-}11$, the quadrilateral formed by the midpoints is a parallelogram.
• Since the diagonals are perpendicular, according to Theorem $6\text{-}16$, the parallelogram is a rhombus.