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In the regular octagon above, what is the value of central angle x?

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In the regular octagon above, what is the value of central angle x?  [#permalink]

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New post 12 May 2015, 12:36
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A
B
C
D
E

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  15% (low)

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In the regular octagon above, what is the value of central angle x?

A 67.5°
B 120°
C 135°
D 145°
E 150°

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Re: In the regular octagon above, what is the value of central angle x?  [#permalink]

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New post 12 May 2015, 23:07
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We will use the symmetry property of a regular polygon to solve this question.

Image

If we observe closely we can see that AE divides the octagon into two equal/symmetrical parts. Similarly, BF divides the octagon into two equal parts. Thus, these two diagonals will bisect the interior angles A & B respectively.

We know the interior angle of a regular polygon can be calculated as \(\frac{(n-2)*180}{n}\). Interior angle of a regular octagon = \(\frac{6*180}{8}\) = 135°. Thus ∠OAB = ∠ OBA = 67.5°.

The central angle X is an exterior angle to the triangle AOB. Thus we can write ∠X = ∠OAB + ∠ OBA = 135°.

Thus the answer is option C

Hope it helps!

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Harsh
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Re: In the regular octagon above, what is the value of central angle x?  [#permalink]

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New post 12 May 2015, 22:52
Notice this quick fact: by drawing lines from various verticies through the center I can split the octagon into 8 equal triangles. The top half of the octagon where the X is located would contain 4 equal triangles that would have the same "inner angle" (I'm calling the angle that touches the center of the octagon the "inner angle" here) that when added together would equal the line that angle X is a portion of. Since a line has 180 degrees and X makes up 3 of these inner angles and we have one inner angle that is not a part of X (the inner angle of the top right triangle shown in the figure) take 1/4 of 180 = 45 and subtract that from 180.

180 - 45 = 135 = Ans C
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Re: In the regular octagon above, what is the value of central angle x?  [#permalink]

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New post 30 Jul 2017, 19:51
Can this logic be applied here. That the inscribed angle is half the central angle?
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In the regular octagon above, what is the value of central angle x?  [#permalink]

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New post 30 Jul 2017, 20:29
reto wrote:
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In the regular octagon above, what is the value of central angle x?

A 67.5°
B 120°
C 135°
D 145°
E 150°


each of the two smaller central angles=360°/8=45°
because angle x is supplementary to 45°,
angle x=180°-45°=135°
C
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In the regular octagon above, what is the value of central angle x?   [#permalink] 30 Jul 2017, 20:29
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