Last visit was: 08 Jun 2026, 06:28 It is currently 08 Jun 2026, 06:28
Home
Messages
MBA Fair
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
805+ (Hard)|   Geometry|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 08 Jun 2026
Posts: 111,139
Own Kudos:
819,517
 [60]
Given Kudos: 106,719
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 111,139
Kudos: 819,517
 [60]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD Updated on: 20 Feb 2019, 03:25
Originally posted by Bunuel on 21 May 2017, 09:43.
Last edited by Bunuel on 20 Feb 2019, 03:25, edited 1 time in total.
Updated.
3
Kudos
Add Kudos
57
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

37% (02:19) correct 63% (02:27) wrong based on 282 sessions

History

Date
Time
Result
Not Attempted Yet
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4


Show Spoiler
Attachment:
main-qimg-5456f088c373424b9e0b766a7449d264.png
main-qimg-5456f088c373424b9e0b766a7449d264.png [ 19.74 KiB | Viewed 175407 times ]
ShowHide Answer
Official Answer
B
Most Helpful Reply
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 05 Jun 2026
Posts: 16,491
Own Kudos:
79,859
 [26]
Given Kudos: 485
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,491
Kudos: 79,859
 [26]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 05:17
13
Kudos
Add Kudos
13
Bookmarks
Bookmark this Post
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4

Show Spoiler
Attachment:
The attachment main-qimg-5456f088c373424b9e0b766a7449d264.png is no longer available
Show more

This question can be solved using the concept of ratio of intercepts on transversals is the same. So if we have have two transversals on 3 parallel lines, the ratio of their intercepts will be the same.

Imagine a line parallel to BF passing through D. Say, there are another two lines parallel to these two lines passing through A and C respectively.
So we have 4 parallel lines and two transversals AD and AC.

Attachment:
Temp2.jpeg
Temp2.jpeg [ 44.72 KiB | Viewed 132560 times ]

Considering 3 parallel lines: the dashed line, BF and the red line,
AE/ED = AF/FP = 1/1

Considering 3 parallel lines: BF, the red line and the yellow line,
BD/DC = FP/PC = 1/1

From these, we get AF = FP = PC and hence AF:FC = 1:2
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 31 Oct 2025
Posts: 6,733
Own Kudos:
36,720
 [5]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,733
Kudos: 36,720
 [5]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 29 Apr 2018, 06:58
1
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4


Show Spoiler
Attachment:
main-qimg-5456f088c373424b9e0b766a7449d264.png
Show more


VERY tricky question. I'd say it's skirting the 800 level of difficulty!

Here's a solution with some diagrams.

First, we'll add the given information.


Next, from point D let's add a line that's PARALLEL to BF


This addition of a parallel line allows us to see a variety of equivalent angles (and similar triangles!)



First, let's examine these two similar triangles.

Since side AD (on the blue triangle) is twice the length of side AE (on the red triangle), we can conclude that the blue triangle is twice as large as the red triangle.
So, side AP is twice the length of side AF.
This tells us that side PF = side AF


Now let's examine these two similar triangles.

Since side BC (on the purple triangle) is twice the length of side DC (on the green triangle), we can conclude that the purple triangle is twice as large as the green triangle.
So, side FC is twice the length of side PC.
This tells us that side PF = side PC

At this point, we know that side PF = side AF and side PF = side PC
Since both equations are set equal to PF, we can conclude that AF = FP = PC

So, the ratio of AF : FC = 1:2

Answer: B

Cheers,
Brent
General Discussion
avatar
laxpro2001
avatar
Joined: 30 Mar 2017
Last visit: 28 Sep 2018
Posts: 29
Own Kudos:
28
Given Kudos: 47
Location: United States (FL)
Posts: 29
Kudos: 28
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 21 May 2017, 12:27
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4


Show Spoiler
Attachment:
main-qimg-5456f088c373424b9e0b766a7449d264.png
Show more


This is a tough one. I guessed - the only progress I was able to make is that BD and DC are equal. I thought AD bisects angle A but I don't see how I can use that to help me and I'm not sure I can make that conclusion even though it makes sense given that BD = DC because each point on the bisector is equidistant from each the sides of the angle.

We also know that AE = ED because E is the midpoint of AD but I don't see how we can use that with what we know about BF to progress. Would love it if someone could provide a good explanation.
User avatar
MaverickTone
User avatar
Joined: 20 May 2017
Last visit: 01 Feb 2018
Posts: 8
Own Kudos:
1
Given Kudos: 2
Posts: 8
Kudos: 1
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 21 May 2017, 13:33
Kudos
Add Kudos
Bookmarks
Bookmark this Post
G is on AC so as EG // BC,

-> GC = 3 FG, also DG = GH.

-> AF = 2FG

-> AF = 1/2 FC

That was my train of thought. Normally when I see mid point I always think of drawing parallels or flipping objects.


Since EG // DC => G is the midpoint of AC. or AG = GC and, EG = 1/2 DC = 1/4 BC.

Also EG // BC => EF = 1/4 BC => EF = 1/3 EB => FG = 1/3 GC

Also, G is mid point of AC, D is mid point of BC => GD // AB => GC = AG.

So AF + FG = GC and FG = 1/3 GC => Af = 2/3 GC = 2/3 AG.

FC = FG + GC = FG + AG = 1/2 AF + 3/2 AF = 2 AF.
Attachments

geometry.png
geometry.png [ 34.55 KiB | Viewed 130463 times ]

avatar
laxpro2001
avatar
Joined: 30 Mar 2017
Last visit: 28 Sep 2018
Posts: 29
Own Kudos:
28
Given Kudos: 47
Location: United States (FL)
Posts: 29
Kudos: 28
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 02:54
Kudos
Add Kudos
Bookmarks
Bookmark this Post
MaverickTone
G is on AC so as EG // BC,

-> GC = 3 FG, also DG = GH.

-> AF = 2FG

-> AF = 1/2 FC

That was my train of thought. Normally when I see mid point I always think of drawing parallels or flipping objects.


Since EG // DC => G is the midpoint of AC. or AG = GC and, EG = 1/2 DC = 1/4 BC.

Also EG // BC => EF = 1/4 BC => EF = 1/3 EB => FG = 1/3 GC

Also, G is mid point of AC, D is mid point of BC => GD // AB => GC = AG.

So AF + FG = GC and FG = 1/3 GC => Af = 2/3 GC = 2/3 AG.

FC = FG + GC = FG + AG = 1/2 AF + 3/2 AF = 2 AF.
Show more


-> GC = 3 FG, also DG = GH
Where is H? I assume that was a typo but I cannot figure out which letter you meant.

And how do you know GC = 3FG?
User avatar
Sahilkm
User avatar
Joined: 02 Feb 2017
Last visit: 25 Dec 2020
Posts: 30
Own Kudos:
5
 [3]
Given Kudos: 1
Posts: 30
Kudos: 5
 [3]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 03:58
3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4


Show Spoiler
Attachment:
main-qimg-5456f088c373424b9e0b766a7449d264.png
Show more

Hpw do I explain this without figure. Okay, let me try:

Consider a point, X, on AC somewhere between F and C. The point X should be such that DX is parallel to BF.
Now, since BF || DX so, CX=XF (parallel base theorem, since BD=DC)
Also, AF= FX (parallel base theorem, since AE=ED)

So CX=XF=AF

Hence, AF: FC = 1:2
User avatar
MaverickTone
User avatar
Joined: 20 May 2017
Last visit: 01 Feb 2018
Posts: 8
Own Kudos:
1
 [1]
Given Kudos: 2
Posts: 8
Kudos: 1
 [1]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 13:07
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
laxpro2001
MaverickTone
G is on AC so as EG // BC,

-> GC = 3 FG, also DG = GH.

-> AF = 2FG

-> AF = 1/2 FC

That was my train of thought. Normally when I see mid point I always think of drawing parallels or flipping objects.


Since EG // DC => G is the midpoint of AC. or AG = GC and, EG = 1/2 DC = 1/4 BC.

Also EG // BC => EF = 1/4 BC => EF = 1/3 EB => FG = 1/3 GC

Also, G is mid point of AC, D is mid point of BC => GD // AB => GC = AG.

So AF + FG = GC and FG = 1/3 GC => Af = 2/3 GC = 2/3 AG.

FC = FG + GC = FG + AG = 1/2 AF + 3/2 AF = 2 AF.


-> GC = 3 FG, also DG = GH
Where is H? I assume that was a typo but I cannot figure out which letter you meant.

And how do you know GC = 3FG?
Show more

Disregard the H, it was a typo.

Anyway, CG = 3 FG is because FG // BC, and the ratio between FG and BC is the same as FG and FC. FC = 4 FG so GC = 3FG.

Now I realize my explanation is a bit messy, but I hope it helps.
avatar
Anurag Nayak
avatar
Joined: 07 Mar 2017
Last visit: 19 Jul 2017
Posts: 1
Given Kudos: 9
Posts: 1
Kudos: 0
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 02 Jun 2017, 14:17
Kudos
Add Kudos
Bookmarks
Bookmark this Post
As per the Menelaus Theorem,
(BC * ED * AF)/(BD*AE*FC)=1,

Assume BD=CD=x ; AE=ED=y ;
So, AF/FC=1/2.
avatar
laxpro2001
avatar
Joined: 30 Mar 2017
Last visit: 28 Sep 2018
Posts: 29
Own Kudos:
28
Given Kudos: 47
Location: United States (FL)
Posts: 29
Kudos: 28
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 04 Jun 2017, 07:55
Kudos
Add Kudos
Bookmarks
Bookmark this Post
VeritasPrepKarishma
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4

Show Spoiler
Attachment:
main-qimg-5456f088c373424b9e0b766a7449d264.png

This question can be solved using the concept of ratio of intercepts on transversals is the same. So if we have have two transversals on 3 parallel lines, the ratio of their intercepts will be the same.

Imagine a line parallel to BF passing through D. Say, there are another two lines parallel to these two lines passing through A and C respectively.
So we have 4 parallel lines and two transversals AD and AC.

Attachment:
Temp2.jpeg

Considering 3 parallel lines: the dashed line, BF and the red line,
AE/ED = AF/FP = 1/1

Considering 3 parallel lines: BF, the red line and the yellow line,
BD/DC = FP/PC = 1/1

From these, we get AF = FP = PC and hence AF:FC = 1:2
Show more

Bunuel
I can't seem to find any good resources that help teach this concept - do you know of any?
User avatar
shashankism
User avatar
Joined: 13 Mar 2017
Last visit: 07 Jun 2026
Posts: 608
Own Kudos:
715
 [4]
Given Kudos: 88
Affiliations: IIT Dhanbad
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE:Engineering (Energy)
Posts: 608
Kudos: 715
 [4]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 04 Jun 2017, 08:19
3
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4


Show Spoiler
Attachment:
The attachment main-qimg-5456f088c373424b9e0b766a7449d264.png is no longer available
Show more

Lets draw a line DG parallel to BF.
Now triangle GCD ~ triangle FCB
-> GC/GF = CD/BD = 1 (Since D is the midpoint of BC)
-> GC = GF

Also triangle AFE ~ AGD
-> AF/GF = AE/ED =1 (Since E is the midpoint of AD)
-> AF = GF

--> AF = GF = GC

So AF / FC = 1/2

Show Spoiler
Answer B

Attachments

main-qimg-5456f088c373424b9e0b766a7449d264.png
main-qimg-5456f088c373424b9e0b766a7449d264.png [ 15.84 KiB | Viewed 128592 times ]

User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 05 Jun 2026
Posts: 16,491
Own Kudos:
79,859
 [4]
Given Kudos: 485
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,491
Kudos: 79,859
 [4]
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 05 Jun 2017, 21:01
3
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
laxpro2001
VeritasPrepKarishma
Bunuel
Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4

Show Spoiler
Attachment:
main-qimg-5456f088c373424b9e0b766a7449d264.png

This question can be solved using the concept of ratio of intercepts on transversals is the same. So if we have have two transversals on 3 parallel lines, the ratio of their intercepts will be the same.

Imagine a line parallel to BF passing through D. Say, there are another two lines parallel to these two lines passing through A and C respectively.
So we have 4 parallel lines and two transversals AD and AC.

Attachment:
Temp2.jpeg

Considering 3 parallel lines: the dashed line, BF and the red line,
AE/ED = AF/FP = 1/1

Considering 3 parallel lines: BF, the red line and the yellow line,
BD/DC = FP/PC = 1/1

From these, we get AF = FP = PC and hence AF:FC = 1:2

Bunuel
I can't seem to find any good resources that help teach this concept - do you know of any?
Show more

Check this out: https://www.youtube.com/watch?v=y5yxL24sRHI
User avatar
Crytiocanalyst
User avatar
Joined: 16 Jun 2021
Last visit: 27 May 2023
Posts: 940
Own Kudos:
214
Given Kudos: 309
Posts: 940
Kudos: 214
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 27 Aug 2021, 06:24
Kudos
Add Kudos
Bookmarks
Bookmark this Post
[quote="Bunuel"]Figure not drawn to scale.

In triangle ABC, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF : FC ?

(A) 1:1

(B) 1:2

(C) 1:3

(D) 2:3

(E) 3:4

it's pretty hard to showcase to show without diagram

when we draw line parallel to BF through D we get 2 similar triangles and when we compare and add all the sides

we get AF/FC = 1:2
Therefore IMO B
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 39,190
Own Kudos:
1,129
Posts: 39,190
Kudos: 1,129
In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 10 Apr 2023, 22:23
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.
Moderator:
Bunuel
Math Expert
111139 posts