Practice GMAT Data Sufficiency Question

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Figure 1
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
Correct Answer: D
  1. Even though lines L and M look parallel and angle BAC looks like a right angle, you cannot make these assumptions.
  2. The formula for the area of a triangle is .5bh
  3. Evaluate Statement (1) alone.
    1. Since EC = 2AC, EA = CA, EC = 2(6) = 12 and line AB is an angle bisector of angle EBC. This means that angle ABC = angle ABE. Since we know that angle ABC = 30, we know that angle ABE = 30. Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90.
    2. Since all the interior angles of a triangle must sum to 180:
      angle ABC + angle BCA + angle BAC = 180
      30 + angle BCA + 90 = 180
      angle BCA = 60
    3. Since all the interior angles of a triangle must sum to 180:
      angle BCA + angle ABC + angle ABE + angle AEB = 180
      60 + 30 + 30 + angle AEB = 180
      angle AEB = 60
    4. This means that triangle BCA is an equilateral triangle.
    5. To find the area of triangle BCE, we need the base (= 12 from above) and the height, i.e., line AB. Since we know BC and AC and triangle ABC is a right triangle, we can use the Pythagorean theorem on triangle ABC to find the length of AB.
      62 + (AB)2 = 122
      AB2 = 144 - 36 = 108
      AB = 1081/2
    6. Area = .5bh
      Area = .5(12)(1081/2) = 6*1081/2
    7. Statement (1) is SUFFICIENT
  4. Evaluate Statement (2) alone.
    1. The sum of the interior angles of any triangle must be 180 degrees.
      DCG + GDC + CGD = 180
      60 + 30 + CGD = 180
      CGD = 90
      Triangle CGD is a right triangle.
    2. Using the Pythagorean theorem, DG = 1081/2
      (CG)2 + (DG)2 = (CD)2
      62 + (DG)2 = 122
      DG = 1081/2
    3. At this point, it may be tempting to use DG = 1081/2 as the height of the triangle BCE, assuming that lines AB and DG are parallel and therefore AB = 1081/2 is the height of triangle BCE. However, we must show two things before we can use AB = 1081/2 as the height of triangle BCE: (1) lines L and M are parallel and (2) AB is the height of triangle BCE (i.e., angle BAC is 90 degrees).
    4. Lines L and M must be parallel since angles FDG and CGD are equal and these two angles are alternate interior angles formed by cutting two lines with a transversal. If two alternate interior angles are equal, we know that the two lines that form the angles (lines L and M) when cut by a transversal (line DG) must be parallel.
    5. Since lines L and M are parallel, DG = the height of triangle BCE = 1081/2. Note that it is not essential to know whether AB is the height of triangle BCE. It is sufficient to know that the height is 1081/2. To reiterate, we know that the height is 8 since the height of BCE is parallel to line DG, which is 1081/2.
    6. Since we know both the height (1081/2) and the base (CE = 12) of triangle BCE, we know that the area is: .5*12*1081/2 = 6*1081/2
    7. Statement (2) alone is SUFFICIENT.
  5. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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