In the figure above ABCD is a rectangle and point K is the c

Question

https://atlanticgmat.com/wp-content/uploads/2014/05/GMAT-Question-of-the-Day-Geometry-Diagram.png

In the figure above ABCD is a rectangle and point K is the center of ABCD. What is the value of x?

(1) KB is equal to MB
(2) M is the midpoint of LC

Hi All,

I thought that you might enjoy a tough Data Sufficiency Geometry question. Good Luck!

qlx · Answer

Hi HerrGrau,

First of all thank you for this tough question

Couldn't establish how B is sufficient , don't you think its high time you provided the solution?

Anyone else ?

Fanja91 · Answer

I'd love an answer to this one as well. That the first part didn't help me I could rule out as the length of the rectangle would affect the value of x, and that's not given. For the second part I'm still dumbstruck.

AtlanticGMAT · Answer

Hi there,

Thank you for participating. Here goes the solution. The easiest way to solve this is by re-drawing the shape a few different ways. For statement 1 you'll notice that you can get x to change. So that statement must be insufficient. For statement 2 you'll get the same of affect although to a lesser extent as the angles flanking x will absorb some of the changes.

Put the statements together and the half rectangle becomes a square. This forces symmetry on the three angles.

For a more in depth explanation you might want to check out this  GMAT Question of the Day  post.

Let me know if you have any questions!

A.

pretzel · Answer

I don't get it either.

AtlanticGMAT · Answer

Hi,

Thanks for participating! My analysis on statement (2) was wrong. I'm not sure why I didn't see this but if you make the rectangle very tall and skinny and x will change. Statement (2) is insufficient. You can also use this process of distortion with Official Guide DS 149.

Happy Studies,

A.

AtlanticGMAT · Answer

Hi All, (updated this for accuracy)

Any questions on this?

Happy Studies,

A.

AtlanticGMAT · Answer

I would love to see an alternate solution to this one. Any geometry wizards out there?

A.

Icerockboom · Answer

(1) easy to see x changes when the measures of the rectangular change.
(2) x changes when the measures of the rectangular change also.
Says, LM = MC = 1/4 DC.
Call the angle of MBC = y. Then tany = MC/BC, tan(x+y)=DC/BC => tany/tan(x+y)=1/4 (m)
tan(x+y)=(tanx+tany)/(1-tanx*tany)
(m)=>tanx =  /(4tany+1) which is not a constant (as tany is not a constant). Then x is not a constant!

(1) & (2) is sufficient.
look at triangle DKL and BMC, DK=BM(=KB), KL/BC=MC/DL=1/2, DLK = MCB = 90 
=> triangle DKL = triangle BMC 
=> DL=BC 
=> MC = 1/2 BC
=> tany = 1/2 which is a constant. then tanx is a constant, then x is a constant.

AtlanticGMAT · Answer

Hi Icerockboom,

Awesome!! Yes - I meant to check back again because I came to the same conclusion. The answer is C because that forces the shape to be a square and the angle to be a constant. But proving the insufficiency of B is a bit too involved for a GMAT question. You really shouldn't have to think about trigonometry. And distorting the shape which would normally work on a GMAT question isn't so easy to see. Really solid work on that. Kudos!

A.

AtlanticGMAT · Answer

Hi All,

So in addition to the trigonometry you can prove that statement (2) is insufficient by distorting the shape. See the diagram and notice how angle x changes in size. Hope this is helpful!

https://atlanticgmat.com/wp-content/uploads/2014/05/GMAT-Question-of-the-Day-Geometry-Solution-5.png

nitin1negi · Answer

I feel answer is B. 
Given is that we have rectangle of any dimensions. K is centre and KL is perpendicular to DC at L.
We have to find angle x which is made by lines joining KB and BM.
I) This option says KB is equal to MB. This question is answered by assuming we are constructing the figure. Finding KB length is simple as it is the distance between K and B. Lets say it is a cms. So MB length is also a cms. But we do not know where the point M is. It could on the line DC as shown or may not be there. Nothing is given. So Insufficient.
II) This option clearly says M is mid point of LC. Now considering this option, we know where the point M is. Also we know where K is. So simply joining K to B and M to B will give the angle KBM i.e. X. So sufficient.

AtlanticGMAT · Answer

Hi nitin1negi,

Good job on statement 1. Statement 2 is tricky. You are right about the midpoint but that doesn't mean that the value of x will be a constant. If you look at the diagrams in the previous post you can see that we can maintain the midpoints while changing the value of x. If the value of x can change while following the rules of the question then the statement must be insufficient. This process of distorting shapes can be helpful when trying to prove insufficiency.

Happy Studies,

A.

igorgomes · Answer

I don't agree on answer being B.

Let the smaller side A and the other B.

Let the angles on corner where is X be theta and gamma.

With only the statement (2), we can come with the following:

tan(theta) = a/2 / b/2
theta = arctan(a/b)

tan(gamma) = b/4 / a
gamma = arctan(b/4a)

So, x = 90 - arctan(a/b) - arctan(b/4a)

So, the answer depends on a and b values. Can't be B.

AtlanticGMAT · Answer

Hi Igor,

Great work - agreed. The answer can't possibly be B based on your assessment. For those who want to avoid trig you can re-draw the shape in a variety of different ways and see that the size of the angle will change.

Happy Studies,

A.

ssriva2 · Answer

Hey  Bunuel ,

Can you please shed some light on this question?
I am not able to understand how the half rectangle can be square?

bumpbot · Answer

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In the figure above ABCD is a rectangle and point K is the c

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GMAT Data Sufficiency (DS) Questions

In the figure above ABCD is a rectangle and point K is the c

Prep Toolkit

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