Square ABCD is the base of the cube while square EFGH is the

Question

Square ABCD is the base of the cube while square EFGH is the cube's top face such that point E is above point A, point F is above point B etc. What is the distance between the midpoint of edge AB and the midpoint of edge EH if the area of square ABCD is 2?

1/sqrt2
1
sqrt2
sqrt3
2sqrt3

Please explain your answer.

A.  \frac{1}{\sqrt{2}} 
B. 1
C.  \sqrt{2} 
D.  \sqrt{3} 
E.  2\sqrt{3}

m14 q23

Bunuel · Accepted Answer

Square ABCD is the base of the cube while square EFGH is the cube's top facet such that point E is above point A, point F is above point B etc. What is the distance between the midpoint of edge AB and the midpoint of edge EH if the area of square ABCD is 2?

A. \(\frac{1}{\sqrt{2}}\)
B. 1
C. \(\sqrt{2}\)
D. \(\sqrt{3}\)
E. \(2\sqrt{3}\)

Look at the diagram below:

Notice that Z is the midpoint of AD. We need to find the length of line segment XY.

Now, since the area of ABCD is 2 then each edge of the cube equals to \(\sqrt{2}\).

\(XZ=\sqrt{AX^2+AZ^2}=\sqrt{(\frac{\sqrt{2}}{2})^2+(\frac{\sqrt{2}}{2})^2}=1\);
\(XY=\sqrt{XZ^2+YZ^2}=\sqrt{1^2+(\sqrt{2})^2}=\sqrt{3}\).

Answer: D.

Show Spoiler

Attachment:

Cube.png [ 14.44 KiB | Viewed 22066 times ]

GMAT TIGER · Answer

distance from mid point of AB to AD = sqrt   = 1


the distance between the midpoint of edge AB and the midpoint of edge EH = sqrt   = sqrt3.

D.

GK_Gmat · Answer

sqrt 3.

let midpoint of EH = M
let midpoint of AB = N
Drop perpendicular from M to side AD on F
AN = (sqrt 2)/2
AF = (sqrt 2)/2
FN =  ^2 +  ^2 = 1
MN^2 = 1^2 + (sqrt 2)^2 = 3
MN = sqrt 3

asdert · Answer

haha! I guesstimated D.

Midpoint AB=M1
Midpoint EF=M2
Midpoint EH=M3

Segment M1M2=sqrt2, hence, M1M3 had to be bigger than that.

Eliminate A, B, C.

Down to D, E. E = 2*1.7*=3.4, which is more than 2 times the segment between M1-M2. No way. D is the only one 8-)

Now, I'll learn how to solve it properly!

Now, I'll learn how to solve it properly!

actionj · Answer

I used deluxe pythag to solve....

We know that the area of the square is 2, therefore the side = sqrt2 or 2^1/2. We know the midpoints are (2^1/2)/2.

So in reality, we are just finding the main diagonal of a rectangular solid with lengths (2^1/2)/2, (2^1/2)/2, and (2^1/2); apply deluxe pythag theorum.

Find x - Main diagonal
((2^1/2)/2)^2 + ((2^1/2)/2)^2 + (2^1/2)^2 = x^2
(2/2)+(2/2)+2=x^2
1/2+1/2+2=x^2
3=x^2
 3^1/2=x

amar13 · Answer

Why angle Z is the right angle?

Bunuel · Answer

YZ is perpendicular to plane ABCD. Hence any line segment which is on the same plane and originates from Z will also be perpendicular to YZ.

Does this make sense?

jayanthjanardhan · Answer

Bunuel,

Can u elaborate a bit more on how the line is perpendicular?

anik19890 · Answer

abcd looks like rombos and it is not like a square then how <xaz= 90?

ENGRTOMBA2018 · Answer

Planes ABCD and ADHE are perpendicular to each other. As such, any such lines drawn on these 2 mutually perpendicular planes will be perpendicular to each other.

ENGRTOMBA2018 · Answer

ABCD can not be a hombus as it a face of a cube. All 6 faces in a cube are squares.  The 1st few words of the question itself mention that ABCD is a square face of the cube.

Additionally, you are basing your observation on a 3D figure drawn on a 2D surface. This is going to lead to a bit of distortion and hence the square 'looks like' a rhombus. In reality, all 6 faces of a cube are square, by definition.

gmatexam439 · Answer

Hello  Bunuel  /  chetan2u ,

The answer will be different if EH were the diagonal of the square. I got this wrong because of that only. 
Either the diagram should be given with the question or it must be mentioned that EH is not a diagonal. I can place E above A and etc in any manner right?

Please correct me if i am wrong.

Regards

chetan2u · Answer

Square ABCD is the base of the cube while square EFGH is the cube's top facet such that point E is above point A, point F is above point B etc

this specifies that ABCD has EFGH over them in same sequence.
yes if it was not given E is over A, F over B etc, you would be correct..

gmatexam439 · Answer

Hello  chetan2u ,

I meant such a figure. In this E is over A, F over B etc, but still the answer will be different.

Am I missing anything here or is the question flawed?

Regards

gmatexam439 · Answer

Bunuel  /  chetan2u ,

Awaiting your response on my above doubt. Please reply.

Regards

chetan2u · Answer

hi..

when a square is given as ABCD, it would mean the corners would be named accordingly ..
AB
DC
or 
BA
CD..
when you read in clockwise or anticlockwise, it should read ABCD..
in the example given by you, the square is ABDC

GMATinsight · Answer

Please find teh solution as attached

Answer: Option D

bumpbot · Answer

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.

Square ABCD is the base of the cube while square EFGH is the

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Square ABCD is the base of the cube while square EFGH is the

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards