roygush wrote:
in the figure above, if MNOPis a trapzoid and NOPRis a parallelogram, what is the area of triangular regoin MNR?
(1) The are of NOPR is 30
(2) The are of OQP is 5
If someone can please explain.
Thanks.
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:
Attachment:
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 30From 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 5Well, 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 statementsNow, 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
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.