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In the figure above, what is the value of x?

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In the figure above, what is the value of x? [#permalink]

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In the figure above, what is the value of x?

(1) AD = AE = BE = CD = BC
(2) x = y
[Reveal] Spoiler: OA

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Last edited by Bunuel on 26 Mar 2013, 01:22, edited 1 time in total.
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Re: In the figure above, what is the value of x? [#permalink]

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emmak wrote:
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In the figure above, what is the value of x?

(1) AD = AE = BE = CD = BC
(2) x = y


Figure attached
STAT1.

Consider Triangle BCD
sum of all three angles = 180
so, y + b + d + e = 180

Also, BC = CD
=> b = d + e

so, equation becomes
y + d + e + d + e = 180
y + 2d + 2e= 180 ... (1)


Consider traingle BCE
sum of all three angles = 180
so, a + c+ y + e = 180

Also, BC = BE
so, a = c+y
so, equation becomes

y + c + y+ c+ y = 180
3y + 2c = 180 ...(2)

Consider traingle ABC
sum of all three angles = 180
so, x + y + c+ d + e = 180 ...(3)

Consider traingle AEB
AE = EB
so, x = d ...(4)

Consider traingle ADC
AD = DC
so, x = c ...(5)

putting d=x (1) becomes
2x + y + 2e = 180 ...(6)

putting c=x in (2) becomes
2x + 3y = 180 ...(7)

putting c=x and d=x in (3) we have
3x+y+e=180 ...(8)

mutiply (8) by 2 and subtract from (6) we have
4x + y = 180 ...(9)

equating (7) and (9) we have
2x+ 3y = 4x+y
=> 2y = 2x
or x=y
putting x=y in (7) we have
5x= 180
x= 36

So, STAT1 is sufficient

STAT2
STAT2 is not sufficient as we dont have relationship between any other angles.

So, Answer will be A.
Will reply in sometime with a smaller and more geometric solution if i get in sometime!

Hope it helps!
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figure.png [ 14.6 KiB | Viewed 4309 times ]


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Re: In the figure above, what is the value of x? [#permalink]

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New post 02 May 2013, 01:32
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A little simpler way

Statement 1:

Angle BEC = z; As AE = EB = CD ==> we have Angle EBD = Angle ECD =x (Isosceles Triangle)
z = 2x (exterior angle theorem)
As BE = BC ==> z = x+y ==> 2x = x + y; means y =x

For Triangle ABC, Angle ACB + Angle CBA + Angle BAC = 180
(x+y) + (x+y) + x = 180
as x = y ==> x =y = 36
Statement 1 sufficient

Statement 2: clearly insufficient

Ans A
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Re: In the figure above, what is the value of x? [#permalink]

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New post 02 May 2013, 02:45
emmak wrote:
Attachment:
1.gif
In the figure above, what is the value of x?

(1) AD = AE = BE = CD = BC
(2) x = y


Good question and quite a learning...

Out of curiousity, can this sum be solved in less than 2mins that too under "GMAT exam" conditions that too at first sight??
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Re: In the figure above, what is the value of x? [#permalink]

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New post 21 May 2013, 15:54
summer101 wrote:
A little simpler way

Statement 1:

Angle BEC = z; As AE = EB = CD ==> we have Angle EBD = Angle ECD =x (Isosceles Triangle)
z = 2x (exterior angle theorem)
As BE = BC ==> z = x+y ==> 2x = x + y; means y =x

For Triangle ABC, Angle ACB + Angle CBA + Angle BAC = 180
(x+y) + (x+y) + x = 180
as x = y ==> x =y = 36
Statement 1 sufficient

Statement 2: clearly insufficient

Ans A


Could you please elaborate on the highlighted one . As to , how did you arrive at Angle EBD = Angle ECD ??

Thanks,
Jyothi
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Re: In the figure above, what is the value of x? [#permalink]

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Re: In the figure above, what is the value of x? [#permalink]

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New post 26 Nov 2015, 11:53
this is my approach.
we can rewrite the sum of the big triangles as:
180=x+2z
where z = 180-2x.

based on statement 1, we can solve for x.
statement 2 doesn't tell us much.
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In the figure above, what is the value of x? [#permalink]

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New post 27 Nov 2015, 11:19
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Detailed Solution



Step-I: Given Info

The question asks us to find the value of x as indicated in the figure, using the 2 statements given.

Step-II: Interpreting the Question Statement

To find the value of x from the given figure, we have to make use of different concepts of geometry based on the information given in the individual statements.
We will also modify the figure as per our convenience to be able to solve the problems easily.
The modified figure is attached as Pic.png(Attached), with all angles marked.

Step-III: Statement-I

Statement- I gives us the information that AD = AE = BE = CD = BC.

Now from the Pic.png(Attached) and using the information in statement 1, we know

Equation 1: x=t=z (Because AE=BE=AD) (Concept Used: Angles opposite to equal sides are equal)
Equation 2: v=t+s & w=z+y (Because BE=BC=CD) (Concept Used: Angles opposite to equal sides are equal)
Equation 3: w=x+t (Concept Used: Exterior angle of a traingle is equal to the Sum of Interior opposite angles)

Combining the results of the above 3 equations

By equation 2 and 3;

w=z+y=x+t.
From Equation 1 x=z=t, so we have x+y=2*x
This means y=x
Also combining this result with equation 1&2, we get y=x=z=t=s

Now considering Triangle ABC,
We know that Sum of Angles in a triangle is 180 degree,

So, \(x+z+y+t+s = 180\)

Since all these 5 angles are equal to each other, we have 5*x=180 or x=36

Hence, statement 1 is sufficient to answer the question.

Step-IV: Statement-II

Statement- II gives us information that x=y. We dis drive this result from statement 1, but it is not sufficient to answer the question since neither value of x or y is given nor any other condition of the triangle is given to come to a definite value of x.

Hence, statement 2 is not sufficient to answer this question.

Step-V: Combining Statements I & II

Since, we have a unique answer from Statement- I so, we don’t need to combine Statements- I & II.
Hence, the correct answer is Option A

Key Takeaways


1. Use of Property: Exterior angle of a triangle is equal to the sum of the interior opposite angles.
2. Use of Property: Angles opposite to equal sides are equal.
In the figure above, what is the value of x?   [#permalink] 27 Nov 2015, 11:19
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