Question

a regular pentagon is shown
what is the length of the apothem, rounded to the
nearest tenth?
9.4 cm
2.9 cm
3.3 cm
4.9 cm
6.5 cm

Explanation:

Step1: Find the central angle

For a regular pentagon, the central angle \(\theta=\frac{360^{\circ}}{5} = 72^{\circ}\). When we consider the right - triangle formed by the radius (\(r = 8\mathrm{cm}\)), half of the side length (\(s/2=\frac{9.4}{2}=4.7\mathrm{cm}\)) and the apothem (\(a\)), the angle at the center of the pentagon for the right - triangle is \(\alpha=\frac{72^{\circ}}{2}=36^{\circ}\).

Step2: Use trigonometric relation

We know that \(\cos\alpha=\frac{a}{r}\) (where \(r\) is the radius of the circum - circle of the regular pentagon and \(a\) is the apothem). Substituting \(\alpha = 36^{\circ}\) and \(r = 8\mathrm{cm}\), we get \(a=r\cos\alpha\).
Since \(\cos(36^{\circ})\approx0.809\), then \(a = 8\times\cos(36^{\circ})\).
\(a=8\times0.809 = 6.472\approx6.5\mathrm{cm}\)

Answer:

\(6.5\mathrm{cm}\)