Chapter 7.4, Problem 8PSB

a.

To determine

Verify the statement.

Answer to Problem 8PSB

Verified.

Explanation of Solution

Given information:

The perpendicular bisector of a side of a regular pentagon passes through the opposite vertex.

Calculation

The figure as per the statement,

  

The perpendicular bisector of a side of a regular pentagon passes through the opposite vertex $\mathrm{A}$ .

  $\angle \mathrm{F}\mathrm{A}\mathrm{B}=54°,\angle \mathrm{A}\mathrm{B}\mathrm{C}=108°,\angle \mathrm{B}\mathrm{C}\mathrm{F}=108°$

  $\angle \mathrm{A}\mathrm{F}\mathrm{C}=360°,\angle \mathrm{A}\mathrm{F}\mathrm{C}=90°$

Hence, verified.

b.

To determine

Verify the statement.

Answer to Problem 8PSB

Verified.

Explanation of Solution

Given information:

Generalize about the perpendicular bisectors of the sides of regular polygons.

Calculation

The figure as per the statement,

  

If you bisect the interior angles of a regular polygon, the bisectors will always converge at the same point - called the incentre of the polygon.

Circumcenter - the point where the three perpendicular bisectors of the sides meet.

Centroid - the point where the altitudes meet.

In the case of regular polygons these centers are all in the same place. In the case of triangles they are usually in different places. (In equilateral triangles, which are essentially three-sided regular polygons, they are in the same.

Hence, verified.