Question

what type of figure is formed by joining the midpoints of a rectangle (see the figure to the right)? justify your answer. choose the correct answer below. a. the figure is a trapezoid. by sss, △dcf≅△agf≅△adh≅△gbh. b. the figure is a square. by sas, △dcf≅△afh. c. the figure is a rectangle. by sss, △agf≅△agh. d. the figure is a rhombus. by sas, △dcf≅△agf≅△adh≅△gbh.

Explanation:

Step1: Recall properties of quadrilaterals

When we join the mid - points of a rectangle, we use the mid - point theorem for triangles and properties of congruent triangles.

Step2: Prove congruent triangles

Let the rectangle be \(ABCD\) and the mid - points of the sides be joined to form a quadrilateral \(EFGH\). In \(\triangle AEH\) and \(\triangle CFG\), \(AE = CF\) (mid - points of opposite sides of rectangle), \(AH=CG\) (mid - points of opposite sides of rectangle) and \(\angle A=\angle C = 90^{\circ}\). By SAS (Side - Angle - Side) congruence criterion, \(\triangle AEH\cong\triangle CFG\). Similarly, \(\triangle BEF\cong\triangle DGH\).

Step3: Analyze side lengths and angles of the formed quadrilateral

The sides of the quadrilateral formed by joining the mid - points of a rectangle are equal (from congruent triangles). Also, the diagonals of the rectangle are equal. Using vector methods or congruent triangle properties, we can show that the adjacent sides of the new quadrilateral are perpendicular. The quadrilateral formed by joining the mid - points of a rectangle is a rhombus.

Answer:

D. The figure is a rhombus. By SAS \(\triangle ECF\cong\triangle EAH\cong\triangle FBG\cong\triangle GDH\)