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805+ (Hard)|   Geometry|      
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Bunuel
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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.
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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 175808 times ]
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 05:17
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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 132946 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
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 29 Apr 2018, 06:58
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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
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 21 May 2017, 12:27
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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.
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 21 May 2017, 13:33
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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.
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 02:54
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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?
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 03:58
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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
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 22 May 2017, 13:07
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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.
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 02 Jun 2017, 14:17
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As per the Menelaus Theorem,
(BC * ED * AF)/(BD*AE*FC)=1,

Assume BD=CD=x ; AE=ED=y ;
So, AF/FC=1/2.
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 04 Jun 2017, 07:55
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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?
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 04 Jun 2017, 08:19
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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

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main-qimg-5456f088c373424b9e0b766a7449d264.png
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KarishmaB
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 05 Jun 2017, 21:01
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laxpro2001
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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
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 27 Aug 2021, 06:24
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[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
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In triangle ABC, D is the mid-point of BC and E is the mid-point of AD 10 Apr 2023, 22:23
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