Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 10 Mar 2014
Posts: 212

What is the area of the trapezoid shown? [#permalink]
Show Tags
01 Aug 2014, 08:54
Question Stats:
50% (00:59) correct 50% (01:01) wrong based on 307 sessions
HideShow timer Statistics
Attachment:
Trapezoid.JPG [ 30.38 KiB  Viewed 14992 times ]
What is the area of the trapezoid shown? (1) Angle A = 120 degrees (2) The perimeter of trapezoid ABCD = 36.
Official Answer and Stats are available only to registered users. Register/ Login.



Intern
Joined: 30 Jul 2014
Posts: 2

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
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



Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4668

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
01 Aug 2014, 10:56
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/gmatdata ... ncerules/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, 60120.JPG [ 22.17 KiB  Viewed 14910 times ]
By the properties of the 306090 triangle, which are explained here: http://magoosh.com/gmat/2012/thegmats ... 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
_________________
Mike McGarry Magoosh Test Prep
Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)



Intern
Joined: 24 Feb 2013
Posts: 9
Location: United States
WE: Information Technology (Consulting)

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
24 Aug 2014, 19:29
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/gmatdata ... ncerules/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, 60120.JPG By the properties of the 306090 triangle, which are explained here: http://magoosh.com/gmat/2012/thegmats ... 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



Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4668

What is the area of the trapezoid shown? [#permalink]
Show Tags
25 Aug 2014, 10:33
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 [ 12.47 KiB  Viewed 14569 times ]
That's a general, nonisosceles 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 "sameside interior angles" between parallel lines. See: http://magoosh.com/gmat/2013/anglesand ... thegmat/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 leftright 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 ... thegmat/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 leftright angles were equal, so we instantly could deduce that the trapezoid is isosceles. Does all this make sense? Mike
_________________
Mike McGarry Magoosh Test Prep
Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)



Intern
Joined: 24 Feb 2013
Posts: 9
Location: United States
WE: Information Technology (Consulting)

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
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, nonisosceles 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 "sameside interior angles" between parallel lines. See: http://magoosh.com/gmat/2013/anglesand ... thegmat/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 leftright 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 ... thegmat/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 leftright angles were equal, so we instantly could deduce that the trapezoid is isosceles. Does all this make sense? Mike



Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4668

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
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 hyperexplicit, spelling conditions out in words. I hope this helps. Mike
_________________
Mike McGarry Magoosh Test Prep
Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)



Director
Joined: 10 Mar 2013
Posts: 562
Location: Germany
Concentration: Finance, Entrepreneurship
GPA: 3.88
WE: Information Technology (Consulting)

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
08 Dec 2015, 15:50
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 306090 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 306090...so same as above we can calculate the Area Answer D
_________________
When you’re up, your friends know who you are. When you’re down, you know who your friends are.
Share some Kudos, if my posts help you. Thank you !
800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50 GMAT PREP 670 MGMAT CAT 630 KAPLAN CAT 660



NonHuman User
Joined: 09 Sep 2013
Posts: 7009

Re: What is the area of the trapezoid shown? [#permalink]
Show Tags
07 Jun 2018, 09:59
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: What is the area of the trapezoid shown?
[#permalink]
07 Jun 2018, 09:59






