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?
b. choose the correct answer below
a. construct two congruent line segments that perpendicularly bisect each other. the endpoints of these line segments will be the midpoints of the sides of a rectangle
b. construct two congruent line segments that perpendicularly bisect each other. the endpoints of these line segments will be the vertices of a rectangle
c. construct two congruent line segments that bisect each other. the endpoints of these line segments will be the vertices of a rectangle
d. construct two congruent line segments that bisect each other. the endpoints of these line segments will be the midpoints of the sides of a rectangle
c. why are the rectangles not congruent? choose the correct answer below
a. if the diagonals are not the same length, the corresponding sides of the two rectangles will not be congruent
b. if the diagonals are not the same length and if the angles formed by the diagonals are not congruent, the corresponding sides of the two rectangles will not be congruent
c. if the angles formed by the diagonals are not congruent, the corresponding sides of the two rectangles will not be congruent
d. if the angles formed by the diagonals are not right angles, the corresponding sides of the two rectangles will not be congruent

Explanation:

a.

1. Let \(ABCD\) be a quadrilateral where the diagonals \(AC\) and \(BD\) are congruent (\(AC = BD\)) and bisect each other at point \(O\), so \(OA=OC\) and \(OB = OD\).

Since \(OA=OC\), \(OB = OD\) and \(\angle AOB=\angle DOC\) (vertically - opposite angles), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle AOB\cong\triangle DOC\). Similarly, \(\triangle AOD\cong\triangle BOC\).

1. Then \(AB = DC\) and \(AD=BC\), so \(ABCD\) is a parallelogram (a quadrilateral with opposite sides congruent).
2. Also, since \(AC = BD\), \(OA=OC\), \(OB = OD\), in \(\triangle AOB\) and \(\triangle BOC\), \(OA = OB=OC = OD\).

In \(\triangle AOB\), \(\cos\angle AOB=\frac{OA^{2}+OB^{2}-AB^{2}}{2\cdot OA\cdot OB}\), and in \(\triangle BOC\), \(\cos\angle BOC=\frac{OB^{2}+OC^{2}-BC^{2}}{2\cdot OB\cdot OC}\).
Since \(AC = BD\), we can show that \(\angle ABC = 90^{\circ}\). A parallelogram with one right - angle is a rectangle.

b.

We know from part (a) that a quadrilateral with congruent and bisecting diagonals is a rectangle. So, we construct two congruent line segments that bisect each other. The endpoints of these line segments will be the vertices of a rectangle.

c.

The rectangles are not congruent because if the angles formed by the diagonals are not congruent, the corresponding sides of the two rectangles will not be congruent. When the angles between the diagonals of two rectangles with equal - length diagonals are different, the lengths of the sides of the rectangles (which can be expressed in terms of the length of the diagonals and the angles between them) will be different.

Answer:

a. See above proof.
b. C. Construct two congruent line segments that bisect each other. The endpoints of these line segments will be the vertices of a rectangle.
c. C. If the angles formed by the diagonals are not congruent, the corresponding sides of the two rectangles will not be congruent.