Chapter 3.6, Problem 45HP

(a)

To determine

To Construct: The distance between the one vertex and the nonadjacent side.

Explanation of Solution

Given:Irregular convex polygon.

The figure of irregular convex polygon us shown below.

  

Use the compass to draw the arcs which intersect the line $AB$ in two points $MandN$ .

Then

Draw the arc from the point $MandN$ with the same opening and cut them.

Join the point $FtoD.$

  

(b)

To determine

To Justify: The line you constructed is perpendicular to the line chosen.

Answer to Problem 45HP

  $\angle DGA={90}^{°}$

Explanation of Solution

Given:Irregular convex polygon.

  

When we measure the angle. It is seen that

  $\angle DGA={90}^{°}$

Thus

The line is perpendicular.

(c)

To determine

To Justify: The construction.

Answer to Problem 45HP

  $DG\perp AB$

Explanation of Solution

Given:Irregular convex polygon.

  

Join the points $MandN$

  $\begin{array}{l}In\Delta DFMand\Delta DFN\\ DM=DN\\ DF=DF\\ MF=NF\\ thus\\ \Delta DFM\cong \Delta DFN\\ therefore\\ \angle MDF=\angle FDN\\ DM=DNby(CPCT)\\ similarly\\ \Delta DGM\cong \Delta DGN\\ now\\ \angle DGM+\angle DGN=180\\ 2\angle DGM=180\\ \angle DGM=90\\ thus\\ DG\perp AB\end{array}$

Thus

The line is perpendicular.