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705-805 (Hard)|   Geometry|      
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v12345
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Given that ABC is a triangle with D being the midpoint of AC and E a p Updated on: 05 Sep 2021, 04:07
Originally posted by v12345 on 01 Sep 2021, 09:38.
Last edited by v12345 on 05 Sep 2021, 04:07, edited 1 time in total.
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

75% (hard)

Question Stats:

47% (02:28) correct 53% (02:22) wrong based on 51 sessions

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Given that ABC is a triangle with D being the midpoint of AC and E a point on CB such that CE = 2EB. If AE and BD intersect at point F and the area of triangle AFB = 1 unit. What is the area of triangle ABC ?
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A. 2
B. 4
C. 5
D. 8
E. 9
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B
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v12345
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Given that ABC is a triangle with D being the midpoint of AC and E a p 05 Sep 2021, 04:07
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Construct a line joining C and F.
Area [ADF] = [DCF] = x (as base and height are same)
Area [BFE] =z, then [FCE] = 2z
Area of AFB = 1
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Area [ADB] = [DCB]
x + 1 = x + 3z
z = 1/3

Area 2x [AEB] =[ACE]
2+ 2z = 2x + 2z
x = 1

Area ABC = 1 + 2x + 3z = 1 + 2 + 1 = 4
Hence, OA is (B).
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antoniogmat
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Given that ABC is a triangle with D being the midpoint of AC and E a p 06 Sep 2021, 10:41
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v12345
Construct a line joining C and F.
Area [ADF] = [DCF] = x (as base and height are same)
Area [BFE] =z, then [FCE] = 2z
Area of AFB = 1
Attachment:
CaptureAns.PNG

Area [ADB] = [DCB]
x + 1 = x + 3z
z = 1/3

Area 2x [AEB] =[ACE]
2+ 2z = 2x + 2z
x = 1

Area ABC = 1 + 2x + 3z = 1 + 2 + 1 = 4
Hence, OA is (B).
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Thank you for your explanation. Could you please explain how do you conclude that 2x [AEB] = [ACE]?
Many thanks. :thumbsup: :thumbsup:
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Given that ABC is a triangle with D being the midpoint of AC and E a p 06 Sep 2021, 20:50
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antoniogmat

Thank you for your explanation. Could you please explain how do you conclude that 2x [AEB] = [ACE]?
Many thanks. :thumbsup: :thumbsup:
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Yes sure.
It is given CE = 2EB
Triangle [AEB] and [ACE] have same height and base of ACE is twice of AEB.
So the area of [ACE] will be 2 * area of [AEB].

Happy to help.
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Given that ABC is a triangle with D being the midpoint of AC and E a p 07 Sep 2021, 03:05
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Say M is a midpoint of CE => CM = ME = EB

We will have DM // FE; because in △CAE, D and M are the midpoints of 02 sides of a triangle

From DM // FE, in △DMB, E is the midpoint of MB => F is the midpoint of DB => DF = FB

From DF = FB => Area of △AFB (the dark triangle) = Area of △ ADF = 1 => Area of △ADB = 2

From AD = DC => Area of △ADB = Area of △BDC; => Area of ABC = 2 + 2 = 4

Answer: B - 4

P/S: Just want to share my "two-cents" pre-think that helps me solve this question:
When I first read the area of the dark triangle = 1, I thought the question would ask me to find another Triangle that has the same area with the dark triangle, then I can sum these numbers together and it may add up to the area of the △ABC. Because △AFB (dark) is in both △ADB and △AEB, the possibility of △ADF is equal to △AFB (dark) is higher than that of △FEB. Therefore, I was looking for the proof of F is the midpoint of DB.
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