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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
If ABCDEFGH is a regular octagon, what fraction of
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06 May 2020, 10:07
06 May 2020, 10:07
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Question Stats:
53% (00:59) correct
46% (02:19) wrong based on 13 sessions
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If ABCDEFGH is a regular octagon, what fraction of the octagon is shaded?
A) 1/12
B) 1/8
C) 1/6
D) 1/4
E) 3/8
Re: If ABCDEFGH is a regular octagon, what fraction of
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09 May 2020, 17:47
09 May 2020, 17:47
1
If we consider three rectangles; PBGQ, BCFG and CRSF
each rectangle is 1/3 part of the complete image.
shaded portion is approximately half of the rectangular part, YGF is approximately 1/6.
if the shaded portion is 1/6th part of the complete square, then it must be a higher portion of the actual octagon.
above 1/6, we have options choices of 1/4 and 3/8. Clearly, YGF is not 3/8 of the octagon. Ans:D
each rectangle is 1/3 part of the complete image.
shaded portion is approximately half of the rectangular part, YGF is approximately 1/6.
if the shaded portion is 1/6th part of the complete square, then it must be a higher portion of the actual octagon.
above 1/6, we have options choices of 1/4 and 3/8. Clearly, YGF is not 3/8 of the octagon. Ans:D
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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If ABCDEFGH is a regular octagon, what fraction of
[#permalink]
10 May 2020, 05:00
10 May 2020, 05:00
2
GreenlightTestPrep wrote:
If ABCDEFGH is a regular octagon, what fraction of the octagon is shaded?
A) 1/12
B) 1/8
C) 1/6
D) 1/4
E) 3/8
If we focus on rectangle BCFG, we might recognize that the area of the shaded portion is half the area of the rectangle BCFG
How do we know this?
Well, we know that the area of the rectangle = (base)(height)
And we know that the area of the triangle = (base)(height)/2
Since both shapes have the same base and height, it must be the case that the area of the shaded portion is half the area of the rectangle BCFG
With this in mind, let's divide rectangle BCFG into TWO EQUAL rectangles.
From here, we can add two smaller triangles.
Since the area of each small triangle is HALF the area of each small rectangle, we can conclude that the combined areas of the two small triangles is HALF the combined areas of the two small rectangles.
In other words, the combined area of the two shaded triangles is the same as the area of the original triangle (triangle YGF)
At this point, when we add a couple of extra diagonals to the diagram...
.... we can see that the shaded portions account for 2/8 of the entire octagon.
Since 2/8 = 1/4, the correct answer is D.
Cheers,
Brent
