pran21 wrote:
Is the triangle PQR equilateral?
(1) Perpendicular bisector of the base QR is also the angular bisector of the vertical angle P.
(2) S is a point such that triangles PQR and PRS are congruent, and PQRS forms a rhombus.
Dear
pran21,
I'm happy to respond.
What is the source of this problem? It is not quite GMAT-like, only because it depends on the details of geometric notation in their precision ---- something that a high school geometry class might test, but I have see a GMAT question hinge on the geometric notation.
Here are the statements with diagrams:
Attachment:
is the triangle equilateral.JPG [ 31.01 KiB | Viewed 5085 times ]
We can easily dispatch with Statement #1. Under that statement, we know that triangle PQR is isosceles, with PQ = PR. It could be equilateral, but doesn't have to be. This statement by itself is
insufficient.
Now, the crux of the problem: Statement #2.
First of all, the fact that PQRS is a rhombus means PQ = QR = RS = PS. Call that length L. If that's the only information we had, then PR could be any other length, and we would still have two congruent isosceles triangles.
The trick of the problem lies in the exact statement of congruence. The rhombus requirement guarantees that PQR is congruent to PSR (order of corresponding points the same), but the prompt tells us that PQR is congruent to PRS --- a different order of the points.
Saying that triangle PQR is congruent to triangle PRS implies that corresponding angles and sides are congruent:
PQ = PR
QR = RS
PR = PS
Since we already know that PQ = QR = RS = PS = L, these statements are also enough to guarantee that PR = L, so each triangle has three congruent sides --- in other words, it's equilateral. This statement is
sufficient.
A clever question, but again, it hinges on geometric notation. It hinges on the test taker recognizing that the two statements
(a) triangle PQR is congruent to triangle PSR
(b) triangle PQR is congruent to triangle PRS
mean two different things. Yes, in geometry, they technically do mean different things, but once again, I have never seen an official GMAT question that hinges on understanding the fine points of geometric notation. In that sense, I don't think this question is representative of what the GMAT would ask.
Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)