Question

consider the construction of a regular hexagon inscribed in a circle.
which of the following statements is true?
select all that apply.

the radius of the circle is equal to the length of the arc between consecutive vertices of the hexagon.

a central angle formed by two consecutive vertices of the hexagon and the center of the circle would measure 60°.

the sum of the lengths of the sides of the hexagon is equal to the circumference of the circle.

the side lengths of the hexagon are equal to the radius of the circle.

Explanation:

Step1: Analyze the first statement

The length of an arc \( s \) is given by \( s = r\theta \) (where \( \theta \) is in radians). For a regular hexagon, the central angle between consecutive vertices is \( 60^\circ=\frac{\pi}{3} \) radians. So the arc length between consecutive vertices is \( r\times\frac{\pi}{3}\), which is not equal to the radius \( r \) (since \( \frac{\pi}{3}\approx1.047
eq1 \)). So the first statement is false.

Step2: Analyze the second statement

A full circle is \( 360^\circ \). In a regular hexagon, there are 6 equal central angles (between consecutive vertices and the center). So each central angle is \( \frac{360^\circ}{6} = 60^\circ \). So this statement is true.

Step3: Analyze the third statement

In a regular hexagon inscribed in a circle, the side length \( s \) of the hexagon is equal to the radius \( r \) of the circle (we'll prove this in the next step). The perimeter of the hexagon is \( 6s = 6r \). The circumference of the circle is \( 2\pi r\approx6.28r \). Since \( 6r
eq2\pi r \) (because \( 6
eq2\pi \)), the sum of the side lengths (perimeter) is not equal to the circumference. So this statement is false.

Step4: Analyze the fourth statement

Consider a regular hexagon inscribed in a circle. If we connect the center of the circle to two consecutive vertices, we form an isosceles triangle with two sides equal to the radius \( r \) and the included angle \( 60^\circ \) (from step 2). By the law of cosines, the length of the side (between the two vertices) is \( \sqrt{r^2 + r^2 - 2r\times r\times\cos(60^\circ)}=\sqrt{2r^2 - 2r^2\times0.5}=\sqrt{2r^2 - r^2}=\sqrt{r^2}=r \). So the side length of the hexagon is equal to the radius of the circle. This statement is true.

Answer:

B. A central angle formed by two consecutive vertices of the hexagon and the center of the circle would measure \( 60^\circ \).
D. The side lengths of the hexagon are equal to the radius of the circle.