Apothem

Find the ratio between the two perimeters, the two areas, and two corresponding apothems of two regular hexagons whose sides are 2-3/4 inches and 4-1/8 inches.

Correct answer:

r₁ =  2/3 = 2:3
r₂ =  4/9 = 4:9
r₃ =  2/3 = 2:3

Step-by-step explanation:

$$\begin{gathered} a=2\frac{3}{4}=2+\frac{3}{4}=\frac{2\cdot 4+3}{4}=\frac{8+3}{4}=\frac{11}{4}=2\frac{3}{4}=2.75\text{in} \\ b=4\frac{1}{8}=4+\frac{1}{8}=\frac{4\cdot 8+1}{8}=\frac{32+1}{8}=\frac{33}{8}=4\frac{1}{8}=4.125 \\ n=6 \\ r=a/b=\frac{11}{4}/\frac{33}{8}=\frac{11}{4}:\frac{33}{8}=\frac{11}{4}\cdot \frac{8}{33}=\frac{11\cdot 8}{4\cdot 33}=\frac{88}{132}=\frac{2}{3}\doteq 0.6667 \\ {r}_1=r=\frac{2}{3}=\frac{2}{3}\approx 0.6667=2:3 \end{gathered}$$

$r_2=r_1^2=0.6667^2=\frac{4}{9}\approx 0.4444=4:9$

$r_3=r_1=0.6667=\frac{2}{3}\approx 0.6667=2:3$

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