 It is currently 25 Nov 2020, 19:11 ### 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

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. # F is the midpoint of AE, and D is the midpoint of CE. Wh  Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
Founder  Joined: 18 Apr 2015
Posts: 13908
GRE 1: Q160 V160 Followers: 313

Kudos [?]: 3678 , given: 12922

F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
Expert's post 00:00

Question Stats: 18% (02:14) correct 81% (02:24) wrong based on 72 sessions

Attachment: #GREpracticequestion F is the midpoint of AE, and D is the midpoint of CE.jpg [ 20.98 KiB | Viewed 3586 times ]

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

Which of the following statements MUST be true?

Indicate all such statements.

❑ FD is parallel to AC.

❑ The area of triangle DEF equals the area of triangle CDF.

❑ The area of triangle ABF is less than the area of triangle DEF.

❑ Angle AFB equals angle BFC.

❑ The area of triangle ACE equals AC x BF.
[Reveal] Spoiler: OA

_________________ Director Joined: 03 Sep 2017
Posts: 518
Followers: 2

Kudos [?]: 440  , given: 66

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
1
KUDOS
Any hint for this question? Founder  Joined: 18 Apr 2015
Posts: 13908
GRE 1: Q160 V160 Followers: 313

Kudos [?]: 3678  , given: 12922

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
2
KUDOS
Expert's post
A is right

Because DE is $$\frac{1}{2}$$ of CE and FE is $$\frac{1}{2}$$ of AE (i.e., corre­sponding sides have proportional lengths), and angle DEF is shared between /the two triangles, you can see that triangles ACE and FDE must be similar. Similar triangles have the further property that corresponding angles are of equal measure. Thus, for example, angle DFE equals angle CAE, and so FD is parallel to AC.

B is right

Looking at the two smaller triangles in the right half of the figure, you can see that triangle CD F and triangle DEF have collinear and equal “bases” (CD and DE), and share their third vertex (F), which is some fixed distance away from
CE. Because CE is comprised of bases CD and DE of triangles CD F and DEF, re­spectively, the two triangles have the same height. Because CDF and DEF have equal bases and the same height, they must have the same area. (For a similar rea­son, triangles ACF and FCE must have equal areas; more on that later.)

Choice E is right

Because triangles ACF and FCE must have equal areas as indicated above, you can see that the area of triangle ACE must be twice that of triangle ACF. Note that FD is parallel to AC due to (true) choice (A), and BF is perpendicular toFD. Therefore, BF must be perpendicular to AC as well. Put differently, AC can be regarded as the base, and BF the height, of triangle ACF. The area of triangle ACF equals ^ times AC x BF. The area of triangle ACE, which is twice that of ACF, must therefore equal AC x BF.
_________________ Intern Joined: 14 Jun 2018
Posts: 36
Followers: 0

Kudos [?]: 9  , given: 100

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
1
KUDOS
Hello guys, any hint why choices C and D are false?
Founder  Joined: 18 Apr 2015
Posts: 13908
GRE 1: Q160 V160 Followers: 313

Kudos [?]: 3678 , given: 12922

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
Expert's post
if you redraw the figure as follow, then C and D not necessarily are true.

Attachment: triangle.png [ 35.83 KiB | Viewed 4416 times ]

_________________
Intern Joined: 07 Jan 2019
Posts: 30
Followers: 0

Kudos [?]: 2 , given: 0

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
thanks
Intern Joined: 01 Jul 2019
Posts: 1
Followers: 0

Kudos [?]: 0 , given: 0

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
For answer choice E, how do you know triangles ACF and FCE are equal in area ?
Founder  Joined: 18 Apr 2015
Posts: 13908
GRE 1: Q160 V160 Followers: 313

Kudos [?]: 3678 , given: 12922

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
Expert's post
Because CE is comprised of bases CD and DE of triangles CD F and DEF, re­spectively, the two triangles have the same height. Because CDF and DEF have equal bases and the same height, they must have the same area. (For a similar rea­son, triangles ACF and FCE must have equal areas; more on that later.)

Ask if you do need further explanations. Sir

Regards
_________________ Director  Joined: 22 Jun 2019
Posts: 517
Followers: 5

Kudos [?]: 107  , given: 161

Re: F is the midpoint of AE, and D is the midpoint of CE. Wh [#permalink]
1
KUDOS
Carcass wrote:
A is right

Because DE is $$\frac{1}{2}$$ of CE and FE is $$\frac{1}{2}$$ of AE (i.e., corre­sponding sides have proportional lengths), and angle DEF is shared between /the two triangles, you can see that triangles ACE and FDE must be similar. Similar triangles have the further property that corresponding angles are of equal measure. Thus, for example, angle DFE equals angle CAE, and so FD is parallel to AC.

B is right

Looking at the two smaller triangles in the right half of the figure, you can see that triangle CD F and triangle DEF have collinear and equal “bases” (CD and DE), and share their third vertex (F), which is some fixed distance away from
CE. Because CE is comprised of bases CD and DE of triangles CD F and DEF, re­spectively, the two triangles have the same height. Because CDF and DEF have equal bases and the same height, they must have the same area. (For a similar rea­son, triangles ACF and FCE must have equal areas; more on that later.)

Choice E is right

Because triangles ACF and FCE must have equal areas as indicated above, you can see that the area of triangle ACE must be twice that of triangle ACF. Note that FD is parallel to AC due to (true) choice (A), and BF is perpendicular toFD. Therefore, BF must be perpendicular to AC as well. Put differently, AC can be regarded as the base, and BF the height, of triangle ACF. The area of triangle ACF equals ^ times AC x BF. The area of triangle ACE, which is twice that of ACF, must therefore equal AC x BF.

Choice E is right

Because triangles ACF and FCE must have equal areas as indicated above, you can see that the area of triangle ACE must be twice that of triangle ACF. Note that FD is parallel to AC due to (true) choice (A), and BF is perpendicular toFD. Therefore, BF must be perpendicular to AC as well. Put differently, AC can be regarded as the base, and BF the height, of triangle ACF. The area of triangle ACF equals ^ times AC x BF. The area of triangle ACE, which is twice that of ACF, must therefore equal AC x BF.

Edited : (The last line will be---) The area of triangle ACF equals = $$\frac{1}{2}$$ * AC x BF. The area of triangle ACE, which is twice that of ACF, must therefore equal AC x BF.

Extended: ACF = $$\frac{1}{2}$$ * AC x BF
=> 2 *ACF = AC x BF
so, ACE = AC x BF (as we know triangle ACE is twice of triangle ACF)
_________________

New to the GRE, and GRE CLUB Forum?
Posting Rules: QUANTITATIVE | VERBAL

Questions' Banks and Collection:
ETS: ETS Free PowerPrep 1 & 2 All 320 Questions Explanation. | ETS All Official Guides
3rd Party Resource's: All In One Resource's | All Quant Questions Collection | All Verbal Questions Collection | Manhattan 5lb All Questions Collection
Books: All GRE Best Books
Scores: Average GRE Score Required By Universities in the USA
Tests: All Free & Paid Practice Tests | GRE Prep Club Tests
Extra: Permutations, and Combination
Vocab: GRE Vocabulary Re: F is the midpoint of AE, and D is the midpoint of CE. Wh   [#permalink] 09 Dec 2019, 05:17
Display posts from previous: Sort by

# F is the midpoint of AE, and D is the midpoint of CE. Wh  Question banks Downloads My Bookmarks Reviews Important topics  Powered by phpBB © phpBB Group Kindly note that the GRE® test is a registered trademark of the Educational Testing Service®, and this site has neither been reviewed nor endorsed by ETS®.