M18-15 : GMAT Club Tests

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Bunuel

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M18-15 [#permalink]

16 Sep 2014, 01:03

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In rectangle $ABCD$, point $E$ is the intersection of the diagonals. If angle $ABD$ is twice the size of angle $EAD$, what is the measure of angle $CED$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

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Bunuel

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Re M18-15 [#permalink]

16 Sep 2014, 01:03

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Official Solution:

In rectangle $ABCD$, point $E$ is the intersection of the diagonals. If angle $ABD$ is twice the size of angle $EAD$, what is the measure of angle $CED$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

Refer to the diagram below:

Observe that since $\angle BAD = 90$ degrees, then $\angle EAB = 90 - x$ degrees.

Furthermore, since the diagonals of a rectangle are equal in length and bisect each other, we have $AE = BE$. Consequently, $\angle EAB = \angle EBA$. From this, we get $90 - x = 2x$, which results in $x = 30$. Thus, $\angle EAB = \angle EBA = 60$ and $\angle AEB = 60$ degrees.

Lastly, since $\angle AEB = \angle CED$, we can conclude that $\angle CED = 60$ degrees.

Answer: C
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MartyTargetTestPrep

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Re: M18-15 [#permalink]

03 Mar 2015, 05:44

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I think this question is good and helpful.
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Senthil7

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Re: M18-15 [#permalink]

16 Aug 2016, 05:48

I think this is a high-quality question and I agree with explanation.

19941010

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Re: M18-15 [#permalink]

11 Sep 2016, 02:25

Bunuel wrote:

Official Solution:

In rectangle $ABCD$, $E$ is the point of intersection of diagonals. If angle $ABD$ is twice angle $EAD$, what is the value of angle $CED$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

Look at the diagram below:

Notice that since $\angle{BAD}=90$ degrees then $\angle{EAB}=90-x$ degrees.

Next, since diagonals of a rectangle are equal and bisect each other then $AE=BE$. So, $\angle{EAB}=\angle{EBA}$: $90-x=2x$, which gives $x=30$. Thus, $\angle{EAB}=\angle{EBA}=60$ and $\angle {AEB}=60$ degrees.

Finally as $\angle{AEB}=\angle{CED}$ then $\angle{CED}=60$ degrees.

Answer: C

Hi,
Aren't the diagonals of a rectangle an angular bisector ? AS, WE KNOW THAT DE = BE. So, angle a should have each angle = 45 degree ? angle DAE = angle BAE = 45 ?
thanks

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Bunuel

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Re: M18-15 [#permalink]

11 Sep 2016, 03:02

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19941010 wrote:

Bunuel wrote:

Official Solution:

In rectangle $ABCD$, $E$ is the point of intersection of diagonals. If angle $ABD$ is twice angle $EAD$, what is the value of angle $CED$?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 90 degrees
E. 120 degrees

Look at the diagram below:

Notice that since $\angle{BAD}=90$ degrees then $\angle{EAB}=90-x$ degrees.

Next, since diagonals of a rectangle are equal and bisect each other then $AE=BE$. So, $\angle{EAB}=\angle{EBA}$: $90-x=2x$, which gives $x=30$. Thus, $\angle{EAB}=\angle{EBA}=60$ and $\angle {AEB}=60$ degrees.

Finally as $\angle{AEB}=\angle{CED}$ then $\angle{CED}=60$ degrees.

Answer: C

Hi,
Aren't the diagonals of a rectangle an angular bisector ? AS, WE KNOW THAT DE = BE. So, angle a should have each angle = 45 degree ? angle DAE = angle BAE = 45 ?
thanks

The diagonal of a rectangle is a bisector only if the rectangle is a square.
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La1yaMalhotra

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Re: M18-15 [#permalink]

26 Sep 2016, 07:33

Aren't the diagonals of rectangle perpendicular bisector of each other?

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Bunuel

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Re: M18-15 [#permalink]

26 Sep 2016, 07:39

Expert Reply

La1yaMalhotra wrote:

Aren't the diagonals of rectangle perpendicular bisector of each other?

The diagonals of a rectangle always bisect each other but they are perpendicular to each other only if a rectangle is a square.
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Re: M18-15 [#permalink]

21 Jun 2017, 07:41

The answer is option C. Well explained above.
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culsivaji

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Re: M18-15 [#permalink]

22 Aug 2017, 14:46

I think this is a high-quality question and I agree with explanation.

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vishalkss

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Re: M18-15 [#permalink]

04 Nov 2017, 02:22

Can you please explain the angle measure 2x for EBA

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Gladiator59

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Re: M18-15 [#permalink]

04 Mar 2019, 00:35

I think this is a high-quality question and I agree with explanation. Good & helpful.
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Re: M18-15 [#permalink]

18 May 2019, 17:17

Good question.

I think what this brings to light are the rules of diagonals in a rectangle and the difference between the bisector of a rectangle and square.

"Each diagonal of a square is the perpendicular bisector of the other. That is, each cuts the other into two equal parts, and they cross and right angles (90°)"

Whereas, each diagonal of a rectangle bisects each other and divides the rectangle up into two equal parts, but the point at which the diagonals bisect does not create 4 equal angles.

Then, finding out what angle corresponds to what side can allow us to solve 90-x = 2x and so forth.

Thanks Bunuel

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RenanBragion

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Re: M18-15 [#permalink]

10 Jun 2020, 04:02

I think this is a high-quality question and I agree with explanation.

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kri1311

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Re: M18-15 [#permalink]

08 Jun 2021, 06:18

Great points in this question:

1. The diagonal of a rectangle is a bisector only if the rectangle is a square.
2. The diagonals of a rectangle always bisect each other but they are perpendicular to each other only if a rectangle is a square.

I made the mistake of assuming the diagonal as a bisector and selected the D option.

Great explanation with great information.

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Bunuel

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