Question

Draw a quadrilateral of the type named. Join, in order, the midpoints of the sides. What special kind of quadrilateral do you appear to get?

Rhombus

Short answer

The quadrilateral formed by joining the mid-point of sides is a rectangle.

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Step description.

From the midpoint theorem of a triangle, any two opposite sides of the rectangle PQRS are parallel and half of the diagonal.

Now, use the midpoint theorem as follows:

$\begin{array}{l}SP\left|\right|BD;SP=\frac{1}{2}BD\\ RQ\left|\right|BD;RQ=\frac{1}{2}BD\end{array}$

Use the transitive property as follows:

$SP\left|\right|RQ;SP=\frac{1}{2}RQ$

So PQRS is a parallelogram.

Step 3. Step description.

Since, diagonals of a rhombus bisect each other at right angles thus $AC\perp BD$.

Since $SP\left|\right|BD,PQ\left|\right|AC,AC\perp BD$ thus, $SP\perp PQ$.

Therefore $\angle QPS={90}^{\circ }$.

Thus, the parallelogram $PQRS$ is a rectangle.