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Albert1
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$A\,\, regular \,\,nonagon \,\,ABCDEFGHI,\,\,if \,\,\overline{AE}=1$
$find :\overline{AB}+\overline{AC}=?$
$find :\overline{AB}+\overline{AC}=?$
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Calculate $\overline{AB}+\overline{AC}$ of Regular Nonagon ABCDEFGHI
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- Thread starter Albert1
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In summary, a regular nonagon is a polygon with nine equal sides and angles. To calculate the average length of two line segments, you add their lengths and divide by two. The length of a line segment on a regular nonagon can be found by dividing the perimeter by nine. The symbols $\overline{AB}$ and $\overline{AC}$ represent line segments, and their sum ($\overline{AB}+\overline{AC}$) on a regular nonagon can be found by doubling the length of a side.
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lfdahl
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My attempt:
View attachment 6451The irregular pentagon $ABCDE$ has a total interior angle sum of $540^{\circ}$. Therefore, $\angle EAB = \angle AED = 60^{\circ}$.
From the figure, we have ($x = \overline{AB}, \: \: \: y=\overline{AC}$):
\[y = \frac{1}{2\cos 40^{\circ}},\: \: \: x = \frac{y}{2\cos 20^{\circ}} = \frac{1}{4\cos 20^{\circ}\cos 40^{\circ}} \\\\ \\\\ x+y = \frac{2\cos 20^{\circ}+1}{4\cos 20^{\circ}\cos 40^{\circ}}=\frac{2\cos 20^{\circ}+1}{2(\cos 20^{\circ}+\cos 60^{\circ})} = 1.\]
From the figure, we have ($x = \overline{AB}, \: \: \: y=\overline{AC}$):
\[y = \frac{1}{2\cos 40^{\circ}},\: \: \: x = \frac{y}{2\cos 20^{\circ}} = \frac{1}{4\cos 20^{\circ}\cos 40^{\circ}} \\\\ \\\\ x+y = \frac{2\cos 20^{\circ}+1}{4\cos 20^{\circ}\cos 40^{\circ}}=\frac{2\cos 20^{\circ}+1}{2(\cos 20^{\circ}+\cos 60^{\circ})} = 1.\]
Attachments
1. What is a regular nonagon?
A regular nonagon is a polygon with nine sides that are all equal in length and nine angles that are all equal in measure.
2. How do you calculate the average length of two line segments?
To calculate the average length of two line segments, you add the length of the two segments and then divide by two.
3. How do you find the length of a line segment on a regular nonagon?
The length of a line segment on a regular nonagon can be found by dividing the perimeter of the nonagon by the number of sides (nine in this case). This will give you the length of each side, which is also the length of the line segment.
4. What does $\overline{AB}+\overline{AC}$ mean?
The symbols $\overline{AB}$ and $\overline{AC}$ represent line segments. $\overline{AB}$ refers to the line segment connecting points A and B, while $\overline{AC}$ refers to the line segment connecting points A and C. The plus sign (+) indicates that we are finding the sum of the two line segments.
5. How do you calculate $\overline{AB}+\overline{AC}$ of a regular nonagon?
To calculate $\overline{AB}+\overline{AC}$ of a regular nonagon, you first find the length of a side by dividing the perimeter by nine. Then, you add that length to itself twice (to represent the two line segments) to get the total length.
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