Chapter 11.4, Problem 1CYU

To determine

To calculate: The center, radius, an apothem and a central angle of the polygon. Also, to find the measure of the central angle

Answer to Problem 1CYU

Center is $F$.

Radius is $FC\text{ or }FD$.

Apothem is $FG$.

Central angle is $\angle DFC$.

The measure of JK is $\angle DFC={90}^{\circ }$.

Explanation of Solution

Given information: A regular hexagon is inscribed in a circle R.

Calculation:

A square is inscribed in a circle R. Therefore, the radius and the center of the circle are the radius and center of the square.

Thus, the radius and center of the square are same as that of the circle.

That is,

Center is $F$.

Radius is $FC\text{ or }FD$.

It can be observed from the figure that CD is one side of the square and FG is the perpendicular that origins from the center of the square and meets the side of the square. Therefore, the segment $\stackrel{\rightharpoonup }{FG}$ is the apothem.

Apothem is $\stackrel{\rightharpoonup }{FG}$.

The vertex of a central angle lies at the center of the polygon. The central angle is such that the sides of the angle passes through consecutive vertices of the given polygon.

The $\angle DFC$ fulfils all these conditions.

Hence, the central angle is $\angle DFC$.

Calculate the measure of the central angle.

The given polygon is a square. Therefore, total number of sides is 4.

The measure of the central angle can be obtained by dividing ${360}^{\circ }$ by the number of sides, that is, 4 in this case.

  $\begin{array}{c}\angle DFC=\frac{{360}^{\circ }}{4}\\ ={90}^{\circ }\end{array}$

Thus, the measure of $\angle DFC={90}^{\circ }$.