Question

The figure shown is a regular hexagon.A 4 in. BFCEDWhat is the length of the diagonal AC?41/3 in.8 in.12 in.8V3 in.

Answer 1

To begin, we first find the measure of the internal angles of the regular hexagon, using the formula below:

`(n-2)/(n)*180`

Since a hexagon is six-sided, we have that n = 6. So:

`(6-2)/(6)*180=(4)/(6)*180=4*30=120`
`\Rightarrow120^o`

Thus, we have each internal angle to be 120 degrees.

Now, we consider the triangle ABC formed in the figure shown in the question.

- We know that

- Also, we know that all the sides of a regular ploygon are always equal and so AB = BC = 4 in.

Thus, we can redraw the figure, as shown below:

Now, we simply apply the Cosine rule to triangle ABC in order to obtain x, which is the length of side AC.

By cosine rule, we have:

`x^2=4^2+4^2-2(4)\cdot(4)\cdot\cos 120^o`

Simplifying gives:

`x^2=16^{}+16^{}-32\cdot\cos 120^o`
`x^2=32^{}-32\cdot(-(1)/(2))`
`x^2=32^{}-(-16)`
`\begin{gathered} x^2=32^{}+16=48 \\ \Rightarrow x^2=48 \end{gathered}`
`\begin{gathered} \Rightarrow x=\sqrt[\square]{48} \\ \Rightarrow x=\sqrt[]{16*3}=\sqrt[\square]{16}*\sqrt[\square]{3}=4*\sqrt[\square]{3} \\ \Rightarrow x=4\sqrt[]{3}\text{ in.} \end{gathered}`

Therefore, the length of the diagonal AC is:

`4\sqrt[]{3}\text{ in.}`

Correct answer: option A