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?
because corresponding parts of congruent triangles are congruent, it follows that ∠abc≅∠dcb
because these angles are also interior angles on the same side of a transversal, they are supplementary
thus m∠abc = m∠dcb = 90 °. (simplify your answer.)
because abcd has a right angle, therefore abcd is a rectangle.
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

Explanation:

Step1: Prove part (a)

Let \(ABCD\) be a quadrilateral with diagonals \(AC\) and \(BD\) such that \(AC = BD\) and \(AC\) and \(BD\) bisect each other at point \(O\). So \(OA=OC\), \(OB = OD\) and \(AC=BD\). In \(\triangle AOB\) and \(\triangle DOC\), \(OA = OC\), \(\angle AOB=\angle DOC\) (vertically - opposite angles) and \(OB = OD\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle AOB\cong\triangle DOC\). Then \(AB = DC\). Similarly, \(\triangle AOD\cong\triangle BOC\), so \(AD = BC\). So \(ABCD\) is a parallelogram (a quadrilateral with both pairs of opposite sides congruent). In \(\triangle ABC\) and \(\triangle DCB\), \(AB = DC\), \(BC=CB\) and \(AC = BD\). By the Side - Side - Side (SSS) congruence criterion, \(\triangle ABC\cong\triangle DCB\). Since corresponding parts of congruent triangles are congruent, \(\angle ABC\cong\angle DCB\). These angles are also interior angles on the same side of a transversal, so they are supplementary. Thus, \(m\angle ABC=m\angle DCB = 90^{\circ}\). Since \(ABCD\) is a parallelogram with a right - angle, \(ABCD\) is a rectangle.

Step2: Solve part (b)

To construct a rectangle using part (a), we know that if we construct two congruent line segments that bisect each other, the quadrilateral formed by their endpoints is a parallelogram. If the line segments are also perpendicular (perpendicularly bisect each other), the parallelogram is a rectangle. The endpoints of these line segments will be the vertices of a rectangle. So the correct answer is B.

Step3: Solve part (c)

Let the length of the diagonal of the rectangle in part (b) be \(d\). To construct a non - congruent rectangle with the same diagonal length, we can change the angle between the diagonals. In a rectangle, the diagonals are congruent and bisect each other. Let the diagonals \(AC\) and \(BD\) bisect each other at \(O\). For two rectangles with the same diagonal length \(d\), if the angles between the diagonals are different, the side - lengths of the rectangles will be different. Let the diagonals of the rectangles be \(AC = BD=d\). If the angle \(\angle AOB\) in one rectangle is \(\theta_1\) and in another rectangle is \(\theta_2
eq\theta_1\), using the law of cosines in the triangles formed by the half - diagonals and the sides of the rectangle (\(AB^{2}=OA^{2}+OB^{2}-2\cdot OA\cdot OB\cdot\cos\angle AOB\), where \(OA = OB=\frac{d}{2}\)), the side - lengths of the two rectangles will be different, so the rectangles are not congruent.

Answer:

a. See the above - step 1 for proof.
b. 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. We can change the angle between the diagonals. Since the side - lengths of the rectangle are related to the angle between the diagonals (using the law of cosines) when the diagonals are of the same length, different angles between the diagonals result in non - congruent rectangles.