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What is the area of the trapezoid shown?

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What is the area of the trapezoid shown?  [#permalink]

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New post 01 Aug 2014, 08:54
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Attachment:
Trapezoid.JPG
Trapezoid.JPG [ 30.38 KiB | Viewed 16274 times ]
What is the area of the trapezoid shown?

(1) Angle A = 120 degrees
(2) The perimeter of trapezoid ABCD = 36.
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Magoosh GMAT Instructor
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Re: What is the area of the trapezoid shown?  [#permalink]

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New post 01 Aug 2014, 10:56
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PathFinder007 wrote:
What is the area of the trapezoid shown?

(1) Angle A = 120 degrees (2) The perimeter of trapezoid ABCD = 36.

Dear PathFinder007,
I'm happy to respond. :-)

Remember, the BIG question on GMAT Data Sufficiency is not "what is the answer?" but rather, "do we have enough information to determine the answer?" This is very subtle --- the sufficiency question is not, "could you in particular find the answer from the information?"; it's really more: "could the ideal math student, given this information, find the answer?" That's the sufficiency question.

Here's a blog that discusses some implication for DS in Geometry:
http://magoosh.com/gmat/2012/gmat-data- ... nce-rules/

So let's look at this:
Statement #1: if angle A = 120, then angle A = angle B = 120, and angle C = angle D = 60. Every angle is determined, and some lengths are specified, so the size and shape are completely determined. That means, the area is completely determined. We don't need to find it. It's enough to know that it's completely determined. Sufficient.

Statement #2: We know AC = BD = 8, because it's an isosceles trapezoid. If we are given the perimeter, then we also know the length of CD, the fourth side. If all four sides are know, that locks the shape in place, determining all the angles and the size and the shape. Again, this completely determines the area. Sufficient.

Both statements sufficient alone. Answer = (D). We can answer the entire DS question without even bothering about calculating the area.

Now, suppose we had a similar PS question in which we had to find the area of this isosceles trapezoid.
Attachment:
isosceles trapezoid, 60-120.JPG
isosceles trapezoid, 60-120.JPG [ 22.17 KiB | Viewed 16187 times ]

By the properties of the 30-60-90 triangle, which are explained here:
http://magoosh.com/gmat/2012/the-gmats- ... triangles/
we know that CE = FD = 4, and of course EF = 6, making the perimeter 36.
AE = BF = \(4sqrt(3)\)
Area of rectangle EABF = \(24sqrt(3)\)
Area of triangle ACE = area of triangle BDF = \(8sqrt(3)\)
Area of isosceles trapezoid CABD = \(40sqrt(3)\)

Does all this make sense?
Mike :-)
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Re: What is the area of the trapezoid shown?  [#permalink]

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New post 01 Aug 2014, 10:33
Diagram is in the attached file.

Area of trapezoid ABCD = ½ X (b1+b2) X Height= ½ X (6+14) X 4√3 =1/2 X 20 X 4√3 = 40√3 cm

Given that angle CAB=120 and Perimeter of ABCD = 36 cm. So CD = 36 – AC – AB – BD
= 36 – 8 – 6 – 8 =14 cm

So we extend A to E and F. Now Angle CAE = 180 – 120 = 60, angle AEC = 90 and angle ACE = 30.

So Triangle ACF is equilateral triangle. we connect CE

As per Pythagoras theorem, AE square + CE square = AC square

=> 4 square + 4√3 square = 8 square

=> 16 + 48 = 64

So here , AE = 4 cm and CD = 4√3 = Height
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Re: What is the area of the trapezoid shown?  [#permalink]

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New post 24 Aug 2014, 19:29
1
1
Hey Mike,

I was reviewing this question and my answer was A because I did not deduce that trapezoid given is an isosceles trapezoid. Can you please explain how did you deduce that the trapezoid given is an isosceles trapezoid in your explanation?

Thanks,
K

mikemcgarry wrote:
PathFinder007 wrote:
What is the area of the trapezoid shown?

(1) Angle A = 120 degrees (2) The perimeter of trapezoid ABCD = 36.

Dear PathFinder007,
I'm happy to respond. :-)

Remember, the BIG question on GMAT Data Sufficiency is not "what is the answer?" but rather, "do we have enough information to determine the answer?" This is very subtle --- the sufficiency question is not, "could you in particular find the answer from the information?"; it's really more: "could the ideal math student, given this information, find the answer?" That's the sufficiency question.

Here's a blog that discusses some implication for DS in Geometry:
http://magoosh.com/gmat/2012/gmat-data- ... nce-rules/

So let's look at this:
Statement #1: if angle A = 120, then angle A = angle B = 120, and angle C = angle D = 60. Every angle is determined, and some lengths are specified, so the size and shape are completely determined. That means, the area is completely determined. We don't need to find it. It's enough to know that it's completely determined. Sufficient.

Statement #2: We know AC = BD = 8, because it's an isosceles trapezoid. If we are given the perimeter, then we also know the length of CD, the fourth side. If all four sides are know, that locks the shape in place, determining all the angles and the size and the shape. Again, this completely determines the area. Sufficient.

Both statements sufficient alone. Answer = (D). We can answer the entire DS question without even bothering about calculating the area.

Now, suppose we had a similar PS question in which we had to find the area of this isosceles trapezoid.
Attachment:
isosceles trapezoid, 60-120.JPG

By the properties of the 30-60-90 triangle, which are explained here:
http://magoosh.com/gmat/2012/the-gmats- ... triangles/
we know that CE = FD = 4, and of course EF = 6, making the perimeter 36.
AE = BF = \(4sqrt(3)\)
Area of rectangle EABF = \(24sqrt(3)\)
Area of triangle ACE = area of triangle BDF = \(8sqrt(3)\)
Area of isosceles trapezoid CABD = \(40sqrt(3)\)

Does all this make sense?
Mike :-)
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Joined: 28 Dec 2011
Posts: 4664
What is the area of the trapezoid shown?  [#permalink]

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New post 25 Aug 2014, 10:33
2
2
karanb wrote:
Hey Mike,

I was reviewing this question and my answer was A because I did not deduce that trapezoid given is an isosceles trapezoid. Can you please explain how did you deduce that the trapezoid given is an isosceles trapezoid in your explanation?

Thanks,
K

Dear karanb,
I'm happy to respond. :-)

Let's say we have any trapezoid ABCD, with top BC parallel to bottom AD. Let's say the two vertices A & B are on the left, and the two vertices C & D are on the right.
Attachment:
general trapezoid.JPG
general trapezoid.JPG [ 12.47 KiB | Viewed 15843 times ]

That's a general, non-isosceles trapezoid. Now, it's always true, 100% of the time for any trapezoid that
(1) (angle A) + (angle B) = 180
(2) (angle C) + (angle D) = 180
because those are "same-side interior angles" between parallel lines. See:
http://magoosh.com/gmat/2013/angles-and ... -the-gmat/
Notice that, for a general trapezoid, all four angles are different. There are two pairs of supplementary angles, but the four angles can have four different numerical values. That will usually be the case for a trapezoid.

OK, so those facts have to be true for any parallelogram. Now, a parallelogram is very special and symmetrical if it also happens to be true that:
(angle A) = (angle D)
(angle B) = (angle C)
In fact, if either one of those is true, the other has to be true. Having left-right angles equal automatically guarantees that the trapezoid is isosceles and that the two legs have equal length. This is analogous to the way that two equal angles in a triangle will automatically guarantee that the triangle is isosceles and that the opposite sides have equal length -- the Isosceles Triangle Theorem. See:
http://magoosh.com/gmat/2012/isosceles- ... -the-gmat/
In fact, if you think about it, the Isosceles Triangle Theorem is the logical basis of what we can deduce about the isosceles trapezoid.

In the original problem on this page, the diagram indicated that the left-right angles were equal, so we instantly could deduce that the trapezoid is isosceles.

Does all this make sense?
Mike :-)
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Re: What is the area of the trapezoid shown?  [#permalink]

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New post 26 Aug 2014, 20:51
Thanks a lot Mike. This explanation is really helpful. I did not notice in the figure that A and D were marked equal. Would GMAT indicate angle/ line equality in the figure with such tick marks? We have used these notations since high school but I am not sure if these are universally accepted notations.

Thanks again!

mikemcgarry wrote:
karanb wrote:
Hey Mike,

I was reviewing this question and my answer was A because I did not deduce that trapezoid given is an isosceles trapezoid. Can you please explain how did you deduce that the trapezoid given is an isosceles trapezoid in your explanation?

Thanks,
K

Dear karanb,
I'm happy to respond. :-)

Let's say we have any trapezoid ABCD, with top BC parallel to bottom AD. Let's say the two vertices A & B are on the left, and the two vertices C & D are on the right.
Attachment:
general trapezoid.JPG

That's a general, non-isosceles trapezoid. Now, it's always true, 100% of the time for any trapezoid that
(1) (angle A) + (angle B) = 180
(2) (angle C) + (angle D) = 180
because those are "same-side interior angles" between parallel lines. See:
http://magoosh.com/gmat/2013/angles-and ... -the-gmat/
Notice that, for a general trapezoid, all four angles are different. There are two pairs of supplementary angles, but the four angles can have four different numerical values. That will usually be the case for a trapezoid.

OK, so those facts have to be true for any parallelogram. Now, a parallelogram is very special and symmetrical if it also happens to be true that:
(angle A) = (angle D)
(angle B) = (angle C)
In fact, if either one of those is true, the other has to be true. Having left-right angles equal automatically guarantees that the trapezoid is isosceles and that the two legs have equal length. This is analogous to the way that two equal angles in a triangle will automatically guarantee that the triangle is isosceles and that the opposite sides have equal length -- the Isosceles Triangle Theorem. See:
http://magoosh.com/gmat/2012/isosceles- ... -the-gmat/
In fact, if you think about it, the Isosceles Triangle Theorem is the logical basis of what we can deduce about the isosceles trapezoid.

In the original problem on this page, the diagram indicated that the left-right angles were equal, so we instantly could deduce that the trapezoid is isosceles.

Does all this make sense?
Mike :-)
Magoosh GMAT Instructor
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Joined: 28 Dec 2011
Posts: 4664
Re: What is the area of the trapezoid shown?  [#permalink]

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New post 27 Aug 2014, 12:10
karanb wrote:
Thanks a lot Mike. This explanation is really helpful. I did not notice in the figure that A and D were marked equal. Would GMAT indicate angle/ line equality in the figure with such tick marks? We have used these notations since high school but I am not sure if these are universally accepted notations.
Thanks again!

Dear karanb,
Yes, the tick marks are used in many high school math texts, but I don't believe it's as universal as, say, the unequal sign or something like that. I have never seen tick marks on an official GMAT diagram --- the GMAT is usually hyper-explicit, spelling conditions out in words.
I hope this helps.
Mike :-)
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Re: What is the area of the trapezoid shown?  [#permalink]

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New post 08 Dec 2015, 15:50
1
PathFinder007 wrote:
Attachment:
Trapezoid.JPG
What is the area of the trapezoid shown?

(1) Angle A = 120 degrees
(2) The perimeter of trapezoid ABCD = 36.


What we have: according to the picture it's a isosceles trapezoid, so AC=BD=8 and CD=AB+2x (if you draw a perpendicular from AB to CD you have to equal lines from the intersection to C and D)

(1) Just draw to perpendicular lines and you'll get to 30-60-90 Trinalges on each side -> if AC=8 so the shorter side=4 and height can be calculated. So we have 2 bases and a height SUFFICIENT
(2) Perimeter=36 -> again, draw 2 perpendicular lines AB -> CD and you'll get to right triangles: 36=2*8+6+6+2x, x=4 if a short leg is 1/2 from a hypotenuse it's a 30-60-90...so same as above we can calculate the Area
Answer D
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Re: What is the area of the trapezoid shown?  [#permalink]

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