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 $ac = bd$. prove that abcd is a rectangle.
because $overline{ac}$ and $overline{bd}$ bisect each other, abcd is a parallelogram and its opposite sides are congruent and parallel.
due to sss congruence, $\triangle abccong\triangle dcb$.
because
it follows that
all angles of congruent triangles are congruent,
corresponding parts of congruent triangles are congruent.

Explanation:

Step1: Recall parallelogram property

If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. Let \(AC\) and \(BD\) bisect each other at \(E\), so \(ABCD\) is a parallelogram with \(AB = CD\), \(AD=BC\) and \(AB\parallel CD\), \(AD\parallel BC\).

Step2: Apply SSS - congruence

In \(\triangle ABC\) and \(\triangle DCB\), we have \(AB = CD\) (opposite - sides of parallelogram), \(BC=CB\) (common side) and \(AC = BD\) (given). By SSS (Side - Side - Side) congruence criterion, \(\triangle ABC\cong\triangle DCB\).

Step3: Use congruent - triangle property

Since \(\triangle ABC\cong\triangle DCB\), by the property that corresponding parts of congruent triangles are congruent, \(\angle ABC=\angle DCB\). Also, since \(AB\parallel CD\) in parallelogram \(ABCD\), \(\angle ABC+\angle DCB = 180^{\circ}\) (adjacent angles of a parallelogram are supplementary). Let \(\angle ABC=x\) and \(\angle DCB=x\) (because \(\angle ABC=\angle DCB\)), then \(x + x=180^{\circ}\), \(2x = 180^{\circ}\), \(x = 90^{\circ}\). Since one angle of the parallelogram \(ABCD\) is a right - angle, \(ABCD\) is a rectangle.

Answer:

A quadrilateral whose diagonals are congruent and bisect each other is a rectangle.