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655-705 (Hard)|   Geometry|      
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mba757
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In the figure above, if MNOP is a trapezoid and NOPR is a pa 17 Sep 2020, 21:15
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adityapagadala
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
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What is the "ASA rule" this user is stating?
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In the figure above, if MNOP is a trapezoid and NOPR is a pa 06 Oct 2020, 08:34
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ScottTargetTestPrep
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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

Show Spoiler
Attachment:
Capture.JPG

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
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Since NOPR is a parallelogram, angle OPQ and angle NRM are equal in measure. Hi Scott, why are they equal ?
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In the figure above, if MNOP is a trapezoid and NOPR is a pa 15 Oct 2020, 12:33
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ScottTargetTestPrep
roygush

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

Show Spoiler
Attachment:
Capture.JPG

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


Since NOPR is a parallelogram, angle OPQ and angle NRM are equal in measure. Hi Scott, why are they equal ?
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Since NOPR is a parallelogram, NR is parallel to OP. Notice that the side RP is a transversal that crosses NR and OP. Thus, the angles OPQ and NRM are corresponding angles and they are congruent.
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In the figure above, if MNOP is a trapezoid and NOPR is a pa 13 Dec 2020, 06:32
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Parallelogram = opposite sides and opposite angles are equal

Trapezoid = one pair of opposite sides is parallel.

Triangle NMR has 2 equal angles, meaning that the triangle is isosceles. If we can determine the area of OQP, we can determine the area of triangular region MNR.

(1) We can't determine the area of triangular region MNR; INSUFFICIENT.

(2) If the shaded region OQP = 5, we can conclude triangular region NMR = 10. SUFFICIENT.
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parallelogram.png
parallelogram.png [ 10.12 KiB | Viewed 4414 times ]

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In the figure above, if MNOP is a trapezoid and NOPR is a pa 30 Dec 2020, 07:36
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Could somebody explain how is it possible to solve the question if the base and height of parallelogram were known?
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In the figure above, if MNOP is a trapezoid and NOPR is a pa 27 Mar 2021, 01:59
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Solution:

St1 :-The area of NOPR is 30.
There is no information about the triangle that can be derived from the area of the parallelogram.(Insufficient)

St(2):-The area of the shaded region is the area of triangle OPQ=5

Triangle MNR is isosceles

=>Height drawn from vertex N to base MR divides the triangle into two congruent right triangles

&

each of these right triangles will have the same area as triangle OPQ.

=> Therefore, the area triangle MNR is twice as much, or 10. Statement two alone is (Sufficient). (option b)

Devmitra Sen(GMAT Quant Expert)
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In the figure above, if MNOP is a trapezoid and NOPR is a pa 30 Mar 2023, 09:34
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