Question

the area of the regular octagon is approximately 54 cm². what is the length of line segment ab, rounded to the nearest tenth? 3.4 cm 4.8 cm 24 cm 27 cm

Explanation:

Step1: Recall the formula for the area of a regular polygon

The area formula for a regular polygon is \(A=\frac{1}{2}aP\), where \(A\) is the area, \(a\) is the apothem, and \(P\) is the perimeter.
Given \(A = 54\space cm^{2}\) and \(a=4\space cm\).

Step2: Solve for the perimeter \(P\)

Substitute the values into the formula \(54=\frac{1}{2}\times4\times P\).
First, simplify the right - hand side: \(\frac{1}{2}\times4\times P = 2P\).
Then, solve for \(P\): \(P=\frac{54}{2}=27\space cm\).

Step3: Find the length of one side

Since a regular octagon has \(n = 8\) sides and \(P=8s\) (where \(s\) is the length of a side).
We know \(P = 27\space cm\), so \(s=\frac{P}{8}=\frac{27}{8}=3.4\space cm\) (rounded to the nearest tenth).

Answer:

3.4 cm