Bunuel wrote:
If angle ABC is 30 degrees, what is the area of triangle BCE?
(1) Angle CDF is 120 degrees, lines L and M are parallel, and AC = 6, BC = 12, and EC = 2AC
(2) Angle DCG is 60 degrees, angle CDG is 30 degrees, angle FDG = 90, and GC = 6, CD = 12 and EC = 12
Are You Up For the Challenge: 700 Level QuestionsThe answer here is (D). Each statement alone is sufficient. Statement 1 alone is sufficient too because of the given data. Normally, I agree it would not be sufficient.
Statement 1:
First think about this - we know that SAS is a congruency rule but SSA is not. So we cannot make a unique triangle using SSA (two sides and one non-included angle). We can make two such triangles given side-side-angle.
Here is a simple video that explains this:
https://www.youtube.com/watch?v=8jaVoGOwbC0The areas of the two triangles would be different.
Considering ACB triangle in our figure, we know AC, CB and angle CBA. So normally, it would give us two such triangles, one acute and one obtuse (as shown in the video). But the measurements given are such that it will give us a 90 degree triangle and hence only one such triangle is possible.
Using the sine law, we know x/Sin X = y/Sin Y
So AC/Sin 30 = BC/Sin BAC
6/Sin 30 = 12/Sin BAC
Since Sin 30 = 1/2, we get 12 = 12/Sin BAC
Sin BAC = 1 which means angle BAC = 90 degrees
(Normally we would get two values of angle BAC because sin Q = sin (180 - Q). So when Q = 90, we get a single value.
This makes BA the altitude of triangle BEC. We can easily find BA because BAC is a 30-60-90 triangle so BA = 6*sqrt(3)
Area of triangle BEC = (1/2) * Altitude * Base \(= \frac{1}{2} * 6\sqrt{3}*12\)
Sufficient
Statement II.
DG is the perpendicular distance between the parallel lines L and M.
Since CDG is a 30-60-90 triangle, we get that DG = 6*sqrt(3)
Since triangle BEC is between the same two parallel lines, the length of altitude will be the same i.e. 6*sqrt(3).
Area of triangle BEC = (1/2) * Altitude * Base \(= \frac{1}{2} * 6\sqrt{3}*12\)
Sufficient
Note: Sufficiency of statement 1 may not be within GMAT scope.
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Karishma
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