Question

a. prove that a quadrilateral whose diagonals are congruent and bisect each other is a rectangle.
b. explain how to use part (a) and only a compass and straightedge to construct any rectangle.
c. construct another rectangle not congruent to the rectangle in part (b) but whose diagonals are congruent to the diagonals of the rectangle in part (b). why are the rectangles not congruent?
a. let $overline{ac}$ and $overline{bd}$ be two line - segments that bisect each other at e, with $overline{ac}congoverline{bd}$. prove that abcd is a rectangle.
because $overline{ac}$ and $overline{bd}$ bisect each other,
abcd is a trapezoid.
abcd is a rhombus and all of its opposite sides are congruent.
abcd is a kite and its adjacent sides are congruent.
abcd is a parallelogram and its opposite sides are congruent and parallel.

Explanation:

Step1: Recall the property of diagonals bisecting each other

If the diagonals of a quadrilateral bisect each other, by the definition of parallelogram - diagonals of a parallelogram bisect each other, the quadrilateral $ABCD$ is a parallelogram and its opposite sides are congruent and parallel.

Step2: Use the congruency of diagonals

In parallelogram $ABCD$, let $\overline{AC}\cong\overline{BD}$. Consider $\triangle ABC$ and $\triangle DCB$. We know that $AB = DC$ (opposite sides of a parallelogram), $BC=CB$ (common side) and $AC = BD$. By SSS (Side - Side - Side) congruence criterion, $\triangle ABC\cong\triangle DCB$. Then $\angle ABC=\angle DCB$. Since $AB\parallel DC$ in parallelogram $ABCD$, $\angle ABC+\angle DCB = 180^{\circ}$. So, $\angle ABC=\angle DCB = 90^{\circ}$. A parallelogram with one right - angle is a rectangle.

Step3: Construction for part (b)

1. Draw a line segment $AC$.
2. Using a compass, find the mid - point $E$ of $AC$ (construct the perpendicular bisector of $AC$ and mark the mid - point).
3. With the compass centered at $E$, draw an arc with radius $AE$ (or $EC$).
4. Using a straightedge, draw a line through $E$ perpendicular to $AC$. Mark points $B$ and $D$ on the arc and the perpendicular line such that $EB = ED=EA = EC$. Then connect $A$, $B$, $C$ and $D$ to form a rectangle.

Step4: Construction for part (c)

1. Start with the same length of diagonal as in part (b). Let the length of the diagonal be $d$.
2. Draw the diagonal $AC = d$.
3. Find the mid - point $E$ of $AC$.
4. This time, draw a different angle for the perpendicular bisector of $AC$. Mark points $B$ and $D$ on the circle centered at $E$ with radius $\frac{d}{2}$ such that the resulting rectangle has different side - lengths. The rectangles are not congruent because although the diagonals are congruent, the angles between the diagonals are different, resulting in different side - lengths (since in a rectangle $l^{2}+w^{2}=d^{2}$, and different angles between the diagonals give different $l$ and $w$ values where $l$ is the length and $w$ is the width and $d$ is the diagonal length).

Answer:

a. The correct option is: ABCD is a parallelogram and its opposite sides are congruent and parallel.
b. See the construction steps above.
c. See the construction steps above.