What is the area of the trapezoid shown?

Question

Trapezoid.JPG What is the area of the trapezoid shown? (1) Angle A = 120 degrees (2) The perimeter of trapezoid ABCD = 36.

mikemcgarry · Accepted Answer

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: https://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. isosceles trapezoid, 60-120.JPG By the properties of the 30-60-90 triangle, which are explained here: https://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 :-)

mikemcgarry · Answer

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. 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: https://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: https://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 :-)

babulsaha · Answer

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

karanb · Answer

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

karanb · Answer

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!

Thanks again!

mikemcgarry · Answer

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 :-)

BrainLab · Answer

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

ShubhamG · Answer

I am little surprised how people have assumed its an isosceles Trapezium and have arrived at D. No where in the diagram mentions the fact that the two non-parallel sides are equal. If the two sides are indeed equal, I guess it has to be EXPLICITLY mentioned in DS. Even if they want to denote equal angles, I think it should be done explicitly - merely showing two dashes for equal angles - doesnt it mean we are assuming (even if it is the broad consensus in mathematics).

Making a question difficult and set traps by making it ambiguous doesnt mean its a good question.

Making a question difficult and set traps by making it ambiguous doesnt mean its a good question.

LipsaTripathy · Answer

Hi , I have doubt regarding the second choice. Even if the perimeter is given it is no where mentioned that it is and isosceles trapezoid. So how can that be sufficient? Thanks in advance

IanStewart · Answer

I agree with both comments above -- there are a few issues with the presentation of the question. You are meant to "deduce" that pairs of angles inside the trapezoid are equal from the diagram (they use identical angle markings for the interior angles at A and B, and for the angles at C and D). You'll never see an official question that does that, however.

sparshgs97 · Answer

I have still not understood why how is this an isoceles trapezoid. where is it given? A is sufficient but i still dont for option D.

Please explain in simple terms how is BD equal to AC.

goaltop30mba · Answer

For people who are having a hard time with option (2), please notice the given figure/diagram— angle a = angle b, and angle c = angle d, either one indicates that the legs of the trapezoid ie AC and BD are equal. All this will lead to the same measurements as in statement (1), which is sufficient. Therefore statement (2) is sufficient as well.

PS: I haven’t written the entire solutions for both the parts/statements as the experts have already explained both extremely well as they always do. Just helping out the people who are having a hard time specifically with statement (2) of this question

Regards,

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What is the area of the trapezoid shown?

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What is the area of the trapezoid shown?

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