Chapter 5.5, Problem 4ST2

To determine

To identify: the type of the quadrilateral $WXYZ$ .

Answer to Problem 4ST2

Square

Explanation of Solution

Given:

Diagonals $\overline{WY}$ and $\overline{XZ}$ are congruent and are perpendicular bisectors of each other.

Concept Used:

Theorem 5-7: the quadrilateral is a parallelogram if the diagonals of a quadrilateral bisect each other.

Theorem 5-12: The diagonals of a rectangle are congruent.

Theorem 5-13: The diagonals of a rhombus are perpendicular.

It is given that for the quadrilateral $WXYZ$ , diagonals $\overline{WY}$ and $\overline{XZ}$ bisects each other.

Thus, by theorem 5-7, the quadrilateral $WXYZ$ is a parallelogram.

Also, it is given that $\text{diag}\text{. }\overline{WY}\cong \text{diag}\text{. }\overline{XZ}$

So, by theorem 5-12, it can be said that the parallelogram $WXYZ$ is a rectangle.

Also, it is given that diagonals $\overline{WY}$ And

  $\overline{XZ}$ are perpendicular to each other.

Thus, by theorem 5-13, the parallelogram $WXYZ$ is a rhombus.

Now, the parallelogram which is both a rhombus and a rectangle is actually a square.