In the figure above, if MNOP is a trapezoid and NOPR is a pa

Question

https://gmatclub.com/forum/download/file.php?id=27559 In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR? (1) The area of NOPR is 30 (2) The area of the shaded region is 5 Capture.JPG

mikemcgarry · Answer

Dear roygush , I'm happy to help. :-)

First of all, here's a important geometry theorem to know: https://magoosh.com/gmat/2012/isosceles- ... -the-gmat/ Triangle MNR has equal base angles --- therefore, it's isosceles, which means MN = NR. Because NOPR is a parallelogram, NR = OP, so that MN = NR = OP. Here's a version of the graph with a new perpendicular line drawn: parallelogram with isosceles triangle.JPG Because the bases are parallel, OQ = NT --- the distance between two parallel lines is always the same. Statement #1 : The area of NOPR is 30 From that, there's no way to determine the areas inside the triangle MNR. That statement, alone and by itself, is insufficient . Statement #2 : The area of OQP is 5 Well, if the shaded region is 5, we could figure out the pieces of the isosceles triangle. Still, we wouldn't know anything about that central part, NOQR. That would remain unknown. Therefore, this statement, alone and by itself, is insufficient . Combined statements Now, we know the area of NOPR is 30. We know that the two little triangles, MNT and TNR, are congruent to QOP, so each has an area of 5. That gives a total area of 30 + 5 + 5 = 40 to the whole trapezoid. With both statements, we can calculate the area, so combined, the statements are sufficient . Answer = (C) Does all this make sense? Mike :-)

adityapagadala · Answer

IMO B

The question asks only about area of triangle MNR

In the figure given by mick..consider the two triangles NTR and OQP

angle NRT = angle OPQ ( as NR and OP are parallel and equal also)
angle NTR = angle OQP = 90 (altitudes)
=> angle TNR = angle QOP

As two angles and the corresponding side (NR and OP) are equal the two triangles, as per ASA rule,are congruent. Hence both will have same area.

Similarly the triangles MNT and NTR are congruent (ASA rule again). Hence the area of triangle MNR = area of MNT + area of NTR = 5+5 = 10

mikemcgarry · Answer

Dear adityapagadala , You are totally 100% correct. I am embarrassed to say --- I made the most boneheaded of all GMAT mistakes: was so concerned with analyzing the diagram, I lost track of exactly what the prompt was asking and assumed it was asking for the area of the whole. You are perfectly correct --- the answer has to be (B) . Mike :-)

CVarga · Answer

Hi Mike,

Would you mind explaining in detail why statement 1 is insufficient. I lost a lot of time on this part of the problem.

Thanks
-Chris

mikemcgarry · Answer

Dear Chris, I'm happy to help. :-)

The area of a parallelogram is (base)*(height). We know nothing about the slant angle x, so we have no idea how much of the parallelogram the triangle occupies. First of all, we know b*h = 30 from Statement #1, but we don't know if we have a base of 1 and a height of 30, or a base of 30 and a height of 1, or something else. Furthermore, we don't know the slant angle x ----- perhaps x = 89°, nearly vertical, so that the side triangle is just a tall skinny sliver, or maybe x is something like 20° or 30°, so that the base of the triangle (QP) is most of the length of the base of the parallelogram (RP). Even knowing the parallelogram has an area of 30, we could design things consistent with this statement & the diagram so that the triangle OQP has an area of nearly zero or nearly half the parallelogram or anything in between. Does all this make sense? Mike :-)

saggii27 · Answer

hey nice solution.

can you please explain the colored part it seems that NRT is exterior angle of given parallelogram, while OPQ is interior angle of the parallellogram.
how are they equal? i lack basics in these geometrical concepts.
thanks in advance.

mikemcgarry · Answer

Dear Saggii27 , I'm happy to respond. :-)

Because NOPR is a parallelogram, this means NR is parallel to OP. This means that RP is a line that cuts across two parallel lines --- such a line is known as a " transversal ." See this blog for all the angles that must be equal in this situation: https://magoosh.com/gmat/2013/angles-and ... -the-gmat/ Does all this make sense? Mike :-)

DJ1986 · Answer

The official answer is B but I'm not quite understanding why we can't use A to divide the area by 3 and get 10. You can tell from the picture that it fits 3 times....

Dondarrion · Answer

Never make any assumptions about figures - especially on DS

They draw them out of scale on purpose to catch people in this very trap.

aniketm.87@gmail.com · Answer

Image
In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR?

(1) The ares of NOPR is 30
(2) The area of the shaded region is 5

A) area of NOPR is 30, this is a parallelogram. It consists of two congruent triangle.

A right proved by others : triangle OPR is congruent to triangle NTR (or half of triangle NMR)

so basically triangle OPR is one of the two congruent triangles which makes up the parallelogram. Since OPR = NTR.

hence half the area of the parallelogram is the area of the req triangle MNR -----> sufficient

Im sure Im going wrong somewhere but this is what i figured in the test. Can someone please point out where am i going wrong in my logic?

Geometry is my weakest area. Thanks in advance

mikemcgarry · Answer

Dear aniketm.87 , I'm happy to respond. :-)

My friend, either diagonal of a parallelogram will indeed divide the parallelogram into two congruent triangles. Do you understand exactly what the the word " congruent " means? It means totally matching in every way: every side in one triangle is exactly equal to the corresponding side in the other, and every angle in one triangle is exactly equal to the corresponding angle in the other. It's absolutely impossible for two triangles to be congruent if one has a right angle and the other doesn't. In parallelogram NOPR, it's true that triangle OPR is congruent to triangle NOR, but that doesn't help us answer the prompt question at all. Triangle NTR, the right triangle that is half of triangle NMR, is not even pretending to be congruent to either OPR or NOR. Triangle NTR has a right angle, and neither OPR or NOR has a right angle. Does all this make sense? Mike :-)

joondez · Answer

- Shape NOPR is a parallelogram so that means lines NR and OP are parallel. 
- If those lines are parallel, then angle R and angle P are equal. So angle P = x.
- If angle R and angle P are equal then triangle QOP is half of triangle MNR, because triangle MNR base angles are both x. 
- You can imagine a line straight down the middle of triangle MNR to create a perpendicular angle, and since the two base angles are both x then triangle MNR is basically just two of triangle QOP.

1.  Not sufficient  because have no base or height information.
2.  Sufficient  because we know triangle QOP is half or triangle MNR. Since triangle QOP is area 5, then triangle MNR must be area 10.

pra1785 · Answer

"As two angles and the corresponding side (NR and OP) are equal the two triangles, as per ASA rule,are congruent. Hence both will have same area."

Is this theory true ? can someone please confirm this.

mikemcgarry · Answer

Dear pra1785 , I'm happy to respond.

Yes, that's 100% correct.

For more detail, see: GMAT Data Sufficiency: Congruence Rules Let me know if you have further questions after reading that blog. Mike

taha1234 · Answer

adityapagadala   mikemcgarry 
How triangles MNT and NTR are congruent?
You are saying that as per ASA rule they are congruent, but how angle TNM and TNR are equal?

mmeuer · Answer

Why can just assume that the line that seemingly creates a triangle from Q to O actually meets in point O ? I understand how to solve the problem based on the assumption but would it not be wrong to just assume that it creates a triangle?

ScottTargetTestPrep · Answer

Solution:

Since NOPR is a parallelogram, angle OPQ and angle NRM are equal in measure. Furthermore, since triangle MNR is isosceles, the height drawn from vertex N to base MR divides the triangle into two congruent right triangles and each of these right triangles will have the same area as triangle OPQ. Therefore, if we know the area of triangle OPQ, then we can determine the area of triangle MNR.

Statement One Only:
 
The area of NOPR is 30.

Knowing the area of the parallelogram doesn’t give us enough information to determine the area of triangle MNR. Statement one alone is not sufficient.

Statement Two Only:

The area of the shaded region is 5.

The area of the shaded region is the area of triangle OPQ. Therefore, the area triangle MNR is twice as much, or 10. Statement two alone is sufficient.

Answer: B

adkor95 · Answer

Thanks Mike - this makes sense, but how can you see that TR = QP? I can see how the rest of the angles/sides are congruent but not that particular piece.

GMATGuruNY · Answer

Let x=40. Since triangle MNR is isosceles, MNOP is a trapezoid, and NOPR is a parallelogram, the following figure is yielded: MNR and NOPR.png The case above illustrates that triangle MNS = triangle NRS = triangle OPQ. Statement 1; The area of region NOPR is 30 It's possible that NOQR = 25 and that triangle OPQ = 5. In this case, since triangle MNS = triangle NRS = triangle OPQ = 5, triangle MNR = triangle MNS + triangle NRS = 5+5 = 10. It's possible that NOQR = 26 and that triangle OPQ = 4. In this case, since triangle MNS = triangle NRS = triangle OPQ = 4, triangle MNR = triangle MNS + triangle NRS = 4+4 = 8. Since triangle MNR can be different values, INSUFFICIENT. Statement 2: The area of the shaded region is 5. Since triangle MNS = triangle NRS = triangle OPQ = 5, triangle MNR = triangle MNS + triangle NRS = 5+5 = 10. SUFFICIENT. B .

In the figure above, if MNOP is a trapezoid and NOPR is a pa

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