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To inscribe a regular decagon in a circle. |
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To inscribe a regular decagon in a circle. |
| Given: Circle A whose radius is AB. |
| Required: To inscribe a regular decagon in circle A. |
| Procedure: |
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Divide AB into extreme and mean ratio, such that AB/AC =
AC/CB (help) |
| Use
the segment command under the construct menu and construct segment AC. |
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Use the circle by center + radius command under the
construct menu and construct a circle with B as the center and radius
congruent to segment AC. |
Label the intersection of the circle B and circle A, point D. 
BD = AC |
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Continue around circle A laying off segments equal
to BD by repeating the last two steps. |
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Use the segment command under the construct menu and connect the
points on circle A in consecutive pairs. |
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Decagon is complete. |
| Proof: |
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AB/AC = AC/BC, hence AB/BD = BD/BC (substitution AC = BD) |
 ABD
=ABD
(identity) |
 ABD
 DBC
(an of
one equals an
of the other and the including sides are proportional) |
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Hence DBC
is isosceles and DB = DC = CA. |
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Hence ABD
= ADB ;
ABD = BCD
and ADC
=BAD
(Base s
of isosceles
are =.) |
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But, BCD
= BAD +ADC
(exterior
of
= sum of 2 opposite interior s.) |
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BCD
= 2BAD
(substitution) |
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ABD
= 2BAD
(substitution) |
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BDA
= 2BAD
(substitution) |
BAD + 2BAD
+2BAD =
180
(The sum of the s
of a
= 180.) |
|
BAD
= 36
(Division) |
|
arc BD
= 36,
or 1/10 of a circle. BD
is a side of a regular decagon. (An equilateral polygon inscribed in a
circle is regular.) |
| Sketch: On the sketch below, the construction on the left was used to divide AB into extreme and mean ratio. These distances are then used in the construction on the right. What point do you have to drag in order to change the size of the decagon? |
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This page uses JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use. inscribe a regular decagon in a circle |
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