F is the midpoint of AE, and D is the midpoint of CE. Which of the fol

Question

https://gmatclub.com/forum/download/file.php?id=48068 F is the midpoint of AE, and D is the midpoint of CE. Which of the following statements MUST be true? I. FD is parallel to AC. II. The area of triangle DEF equals the area of triangle CDF. III. The area of triangle ABF is less than the area of triangle DEF. A. I only B. II only C. III only D. I and II only E. I, II and III #GREpracticequestion F is the midpoint of AE, and D is the midpoint of CE.jpg

nick1816 · Answer

1. ED/CE= EF/EA=1/2
Hence FD is parallel to AC. Must be true!!

2. As F is the midpoint of AE
Area of ACF= Area of FCE= 1/2(Area of ACE).....(1)

Also ED/CE= EF/EA=1/2 
Area of FED= 1/4(area of ACE)......(2)

From (1) and (2)
The area of triangle DEF equals the area of triangle CDF. Must be true!!!

3. BF is perpendicular to AC, but we have no idea about the location of B on AC or we don't know the AB/BC. 
Hence 3rd statement can be true or false.

abhishek31 · Answer

1. ED/CE= EF/EA=1/2
Hence FD is parallel to AC. Must be true!!

2. As F is the midpoint of AE
Area of ACF= Area of FCE= 1/2(Area of ACE).....(1)

Also ED/CE= EF/EA=1/2 
Area of FED= 1/4(area of ACE)......(2)

From (1) and (2)
The area of triangle DEF equals the area of triangle CDF. Must be true!!!

3. BF is perpendicular to AC, but we have no idea about the location of B on AC or we don't know the AB/BC. 
Hence 3rd statement can be true or false.

Hi, can you please ellaborate on point (1), how did you use the concept of proportion to determine the parallelism between the 2 sides?

nick1816 · Answer

Midpoint theorem- Line joining the midpoints of the two sides of triangle is parallel to the third side.
It's easy to prove this theorem. With SAS you can tell that FED~AEC.
Hence angle DFE=angle CAE, and angle FDE= angle ACE
If corresponding angles are equal, then both lines must be parallel.

Hi, can you please ellaborate on point (1), how did you use the concept of proportion to determine the parallelism between the 2 sides?

CrackverbalGMAT · Answer

There are a couple of triangle concepts tested here in this question.

If a line is drawn such that it connects the midpoints of any two sides of a triangle, it will be parallel to the third side and will be half of the third side.

The above is enough to prove that FD is parallel to AC. Statement I must be true. So, any option/s not containing I can be eliminated. Options B and C go out.

If two triangles have the same base and lie between the same parallel lines, they will have the same area.

Let’s draw a line parallel to EC, through F. Then, triangles DEF and CDF have equal bases (since D is the midpoint of C) and lie between the same parallel lines.

Attachment:

30th May 2019 - Reply 2.JPG [ 17.48 KiB | Viewed 3827 times ]

Clearly, statement II must be true. Option A can now be eliminated. Possible answer options are D and E.

We do not have enough data to conclusively say if area of triangle ABF will always be lesser than area of triangle DEF. This is because of the relative positions of B and D, due to which we do not know the heights of these triangles. So, statement III is not necessarily true. Option E can now be eliminated.

Answer option D has to be the correct answer.

If you know both the concepts stated above, this question should take you a minute, on an average to solve. So, this is actually a moderately difficult question. Also remember that you should be able to make basic constructions to be able to apply the concepts. Solving sufficient number of questions where you are required to make constructions, will help you develop this skill.

Hope this helps!

F is the midpoint of AE, and D is the midpoint of CE. Which of the fol

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F is the midpoint of AE, and D is the midpoint of CE. Which of the fol

Prep Toolkit

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My Rewards