Question

For exercises, 14-18 write paragraph proofs.

Given: parallelogram ABCD; W, X, Y, Z are midpoints of $\overline{\mathrm{AO}}$, $\overline{\mathrm{BO}}$, $\overline{\mathrm{CO}}$and $\overline{\mathrm{DO}}$.

Prove: role="math" localid="1637745814874" $\mathrm{WXYZ}$ is a parallelogram.

Short answer

It is proved that the quadrilateral WXYZ is a parallelogram.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that ABCD is a parallelogram.

In a parallelogram, both diagonals bisect each other.

Therefore, in the parallelogram ABCD, both diagonals AC and BD intersect each other at the point O.

Therefore, $AO=BO$and $CO=DO$.

Step 3. Description of step.

It is also being given that W, X, Y, Z are midpoints of $\overline{AO},\overline{BO},\overline{CO}$and $\overline{DO}$.

As, W is the midpoint of $\overline{AO}$, therefore by using the definition of midpoint it can be said that $AO=2WO$.

As, X is the midpoint of $\overline{BO}$, therefore by using the definition of midpoint it can be said that $BO=2XO$.

As, Y is the midpoint of $\overline{CO}$, therefore by using the definition of midpoint it can be said that $CO=2YO$.

As, Z is the midpoint of $\overline{DO}$, therefore by using the definition of midpoint it can be said that $DO=2ZO$.

Therefore, it can be noticed that:

$\begin{array}{c}AO=BO\\ 2WO=2XO\\ WO=XO\end{array}$

and

$\begin{array}{c}CO=DO\\ 2YO=2ZO\\ YO=ZO\end{array}$

Therefore, it can be seen that the diagonals WY and XZ are bisecting each other at the point O.

Therefore, the quadrilateral WXYZ is a parallelogram.