Question

This figure represents a design found in a glass panel. ABCD is a rectangle with

Answer 1

Answer: Yes

Step-by-step explanation:

Since ABCD is a rectangle, $\angle AXY$, $\angle YBZ$, $\angle WCZ$, and $\angle WDX$ are all right angles, and are thus all congruent because all right angles are congruent. Furthermore, because ABCD is a rectangle, we know that $\overline{AB} \cong \overline{CD}$ and $\overline{AD} \cong \overline{BC}$. Because we are given that X, Y, Z, and W are midpoints, using the fact that halves of congruent segments are congruent, we can conclude that $\overline{AY} \cong \overline{YB} \cong \overline{CW} \cong \overline{WD}$ and that $\overline{AX} \cong \overline{XD} \cong \overline{BZ} \cong \overline{ZC}$. As a result, we can conclude that $\triangle AYX \cong \triangle DXW \cong \triangle CWZ \cong \triangle BYZ$ by SAS, and thus by CPCTC, $\overline{AY} \cong \overline{XW} \cong \overline{ZW} \cong \overline{YZ}$. Therefore, since the quadrilateral formed by the midpoints has four congruent sides, it must be a rhombus.