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655-705 (Hard)|   Geometry|      
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Bunuel
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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segme 22 Apr 2019, 02:14
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

55% (hard)

Question Stats:

63% (02:52) correct 37% (03:36) wrong based on 27 sessions

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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segment AB is extended 2 units beyond B to point E, and F is the intersection of ED and BC. What is the area of BFDG?


(A)133/2
(B) 67
(C) 135/2
(D) 68
(E) 137/2
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C
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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segme 22 Apr 2019, 09:54
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extend DC to a point X parallel to BE and join EX to form a rectangle BCXE.

We have two similar triangles DEA and FEB (since angle DAE=angle FBE=90 and angle E is common for both triangles)
therefore AD/BF=AE/BE implies 30/BF=8/2 which gives BF=7.5

Now area BFDG= area AEXD - area XED - area AGB - area FBE
= 30*8 - 1/2*8*30 - 1/2*6*15 - 1/2*2*7.5
= 135/2
IMO OA is C

You can also solve without extending the lines to X with the value CF(22.5) as we already know BF
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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segme 23 Apr 2019, 07:13
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Can someone please help with this?
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nick1816
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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segme 23 Apr 2019, 07:31
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triangle DCF is similar to triangle EBF
Hence, (DC/CF)= (EB/BF)
we get CF/BF=3 Also we know CF+BF=30
hence CF=22.5
Area of BFDG= Area of ABCD - ARea of DCF - Area of GAB
= (30*6) - (0.5*6*22.5) - (0.5*6*15) = 67.5
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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segme 23 Apr 2019, 08:28
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nick1816
triangle DCF is similar to triangle EBF
Hence, (DC/CF)= (EB/BF)
we get CF/BF=3 Also we know CF+BF=30
hence CF=22.5
Area of BFDG= Area of ABCD - ARea of DCF - Area of GAB
= (30*6) - (0.5*6*22.5) - (0.5*6*15) = 67.5
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Thanks, somehow didn't pay attention to the "We have two similar triangles DEA and FEB (since angle DAE=angle FBE=90 and angle E is common for both triangles)
therefore AD/BF=AE/BE implies 30/BF=8/2 which gives BF=7.5" quoted text from swarup's post. After looking at your reply, checked Swarup's post as well..
Thanks to you and swarup..
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In rectangle ABCD, AB = 6, AD = 30, and G is the midpoint of AD. Segme 12 Oct 2020, 04:19
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