Question

the apothem of a regular hexagon measures 8 cm which are true of the regular hexagon? check all that apply the perimeter of the hexagon is 48 cm the measure of the angle formed by the radius and the apothem is 30° the side length of the hexagon is about 4 6 cm in a regular hexagon, the radius and side length are equal in length the area of the hexagon is about 221 7 square cm

Explanation:

Step1: Analyze the properties of a regular hexagon

A regular hexagon can be divided into six equilateral triangles. The central angle of a regular hexagon is \(\frac{360^{\circ}}{6} = 60^{\circ}\). When we consider the right - triangle formed by the apothem (\(a = 8\) cm), half of the side length (\(s/2\)) and the radius (\(r\)), the angle between the radius and the apothem is \(\frac{60^{\circ}}{2}=30^{\circ}\).
We know that \(\tan30^{\circ}=\frac{s / 2}{a}\). Since \(a = 8\) cm, \(\frac{s}{2}=a\tan30^{\circ}=8\times\frac{\sqrt{3}}{3}\approx4.6\) cm, then \(s=\frac{16\sqrt{3}}{3}\approx9.2\) cm. The perimeter \(P = 6s\approx6\times9.2 = 55.2\) cm.
The area formula of a regular polygon is \(A=\frac{1}{2}aP\). If \(a = 8\) cm and \(P\approx55.2\) cm, then \(A=\frac{1}{2}\times8\times55.2=220.8\approx221.7\) square cm. Also, in a regular hexagon, the radius \(r\) is equal to the side length \(s\) (because the six equilateral triangles formed have side lengths equal to the radius and the side length of the hexagon).

Step2: Check each option

• Option 1: The perimeter of the hexagon is \(48\) cm

If \(s = 8\) cm (incorrect assumption, we found \(s\approx9.2\) cm), \(P = 6s\). If \(s = 8\), \(P=48\), but our calculation using the apothem shows \(s
eq8\), so this is wrong.

• Option 2: The measure of the angle formed by the radius and the apothem is \(30^{\circ}\)

Since the central angle of a regular hexagon is \(60^{\circ}\), and the apothem bisects the central - angle - to - side angle. So the angle between the radius and the apothem is \(\frac{60^{\circ}}{2}=30^{\circ}\), this is correct.

• Option 3: The side length of the hexagon is about \(4.6\) cm

\(\frac{s}{2}=a\tan30^{\circ}\), \(s = 2a\tan30^{\circ}\). If \(a = 8\), \(s=\frac{16\sqrt{3}}{3}\approx9.2\) cm. The value \(4.6\) is half of the side - length (from the right - triangle relation in the hexagon), so this is wrong.

• Option 4: In a regular hexagon, the radius and side length are equal in length

A regular hexagon can be divided into six equilateral triangles. The side of the equilateral triangle is equal to the radius of the circum - circle of the hexagon and also equal to the side length of the hexagon. So this is correct.

• Option 5: The area of the hexagon is about \(221.7\) square cm

Using \(A=\frac{1}{2}aP\), with \(a = 8\) and \(P\approx55.2\), \(A=\frac{1}{2}\times8\times55.2 = 220.8\approx221.7\) (due to rounding in side - length calculation). So this is correct.

Answer:

The measure of the angle formed by the radius and the apothem is \(30^{\circ}\); In a regular hexagon, the radius and side length are equal in length; The area of the hexagon is about \(221.7\) square cm.