Chapter 9.5, Problem 23WE

To determine

To Find: The type of figure the quadrilateral AQPB.

Answer to Problem 23WE

The figure is a rectangle.

Explanation of Solution

Given information:

Equilateral $△ABC$ is inscribed in a circle.

P and Q are midpoints of BC and CA.

Consider the following figure:

  

In equilateral ΔABC each angle = $60°$

Therefore, $\stackrel{⌢}{AC}=120°$

Also, $\stackrel{⌢}{AQ}=\frac{1}{2}\stackrel{⌢}{AC}$

  $\begin{array}{l}\stackrel{⌢}{BAQ}=60°+120°\hfill \\ =180°\hfill \end{array}$

Hence, $\angle BAQ=90°$

Similarly, $\angle BPQ=90°$

As all angles are right angled, we can conclude that the figure is a rectangle.