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# polygon length and area question

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The lengths of the perpendiculars drawn to the sides of a regular hexagon from an interior point are 4, 5, 6, 8, 9, and 10 centimeters. What is the number of centimeters in the length of a side of this hexagon? Express your answer as a common fraction in simplest radical form.

Dec 5, 2018

#1
+22273
+11

The lengths of the perpendiculars drawn to the sides of a regular hexagon from an interior point are 4, 5, 6, 8, 9, and 10 centimeters.

What is the number of centimeters in the length of a side of this hexagon?

$$\begin{array}{|rcll|} \hline 7 = \dfrac{4+10}{2} = \dfrac{5+9}{2}= \dfrac{6+8}{2} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline {\color{red} s}^2 &=& \left({\color{red}\dfrac{s}{2}}\right)^2 +7^2 \\\\ s^2 &=&\dfrac{s^2}{4} + 7^2 \\\\ s^2-\dfrac{s^2}{4} &=& 7^2 \\\\ \dfrac{3}{4} s^2 &=& 7^2 \\\\ s^2 &=& \dfrac{4}{3} \cdot 7^2 \quad & | \quad \text{sqrt both sides}\\\\ s &=& \dfrac{2}{\sqrt{3}} \cdot 7 \\\\ s &=& \dfrac{14}{\sqrt{3}} \\\\ s &=& \dfrac{14}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}} \\\\ \mathbf{s} & \mathbf{=} & \mathbf{\dfrac{14}{3}\cdot \sqrt{3} } \\ \hline \end{array}$$

The number of centimeters in the length of a side of this hexagon is $$\mathbf{\dfrac{14}{3}\cdot \sqrt{3} } \ \text{cm}$$

Dec 6, 2018
#2
+22273
+11

The lengths of the perpendiculars drawn to the sides of a regular hexagon from an interior point are 4, 5, 6, 8, 9, and 10 centimeters.
What is the number of centimeters in the length of a side of this hexagon?

With Aera A:

$$\begin{array}{|rcll|} \hline A &=& \dfrac{s^2\cdot \sin(60^{\circ})}{2} \cdot 6 \quad & | \quad \sin(60^{\circ}) = \dfrac{\sqrt{3}}{2} \\\\ &=& \dfrac{s^2\cdot \dfrac{\sqrt{3}}{2}}{2} \cdot 6 \\\\ \mathbf{A} & \mathbf{=} & \mathbf{\dfrac{3}{2} \sqrt{3}s^2 } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline A &=& \dfrac{s\cdot 4}{2}+ \dfrac{s\cdot 5}{2}+ \dfrac{s\cdot 6}{2}+ \dfrac{s\cdot 8}{2}+ \dfrac{s\cdot 9}{2}+ \dfrac{s\cdot 10}{2} \\\\ &=& \dfrac{1}{2}(4+5+6+8+9+10)s \\\\ &=& \dfrac{1}{2}\cdot 42s \\\\ \mathbf{A} & \mathbf{=} & \mathbf{21s} \\ \hline \end{array}$$

$$\mathbf{s =\ ?}$$

$$\begin{array}{|rcll|} \hline \mathbf{\dfrac{3}{2} \sqrt{3}s^2 } &=& \mathbf{21s} \\\\ \dfrac{3}{2} \sqrt{3}s &=& 21 \\\\ s &=& 21 \dfrac{2}{3\sqrt{3}} \\\\ s &=& \dfrac{14}{\sqrt{3}} \\\\ s &=& \dfrac{14}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}} \\\\ \mathbf{s} & \mathbf{=} & \mathbf{\dfrac{14}{3}\cdot \sqrt{3} } \\ \hline \end{array}$$

The number of centimeters in the length of a side of this hexagon is $$\mathbf{\dfrac{14}{3}\cdot \sqrt{3} } \ \text{cm}$$

Dec 6, 2018