Hi,
I encountered this problem while taking the Platinum GMAT practice test.
Link to Question: https://www.platinumgmat.com/gmat-practi ... on_id=2052
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
In the answer explanation below, 3.1 states that "Since EC = 2AC, EA = CA, EC = 2(6) = 12 and line AB is an angle bisector of angle EBC". But based on the fact that EA=AC=6 and EC = BC = 12, we can only conclude AB is the median of EC right? Unless we atleast know Tri-EBC is isoc (EB=BC) the angle bisector and the median need not be the same.
3.1 also states, "Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90". Not sure how L and M being parallel make AB perpendicular to L and M.
I was a little thrown off by this explanation. It is possible that both the above are true (probably by using the fact that CDF = 120, although I am not able to arrive at that result) - but the explanation seems to be assuming both with.
Any help in pointing out if I missed anything would be greatly appreciated.
Thanks,
===============Answer Explanation=====================
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.
3.1. 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.
3.2. 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
3.3. 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
3.4. This means that triangle BCA is an equilateral triangle.
3.5. 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
3.6. Area = .5bh"¨Area = .5(12)(1081/2) = 6*1081/2
3.7. Statement (1) is SUFFICIENT
4. Evaluate Statement (2) alone.
4.1. 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.
4.2. Using the Pythagorean theorem, DG = 1081/2"¨(CG)2 + (DG)2 = (CD)2"¨62 + (DG)2 = 122"¨DG = 1081/2
4.3. 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).
4.4. 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.
4.5. 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.
4.6. 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
4.7. Statement (2) alone is SUFFICIENT.
5. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.
I encountered this problem while taking the Platinum GMAT practice test.
Link to Question: https://www.platinumgmat.com/gmat-practi ... on_id=2052
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
In the answer explanation below, 3.1 states that "Since EC = 2AC, EA = CA, EC = 2(6) = 12 and line AB is an angle bisector of angle EBC". But based on the fact that EA=AC=6 and EC = BC = 12, we can only conclude AB is the median of EC right? Unless we atleast know Tri-EBC is isoc (EB=BC) the angle bisector and the median need not be the same.
3.1 also states, "Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90". Not sure how L and M being parallel make AB perpendicular to L and M.
I was a little thrown off by this explanation. It is possible that both the above are true (probably by using the fact that CDF = 120, although I am not able to arrive at that result) - but the explanation seems to be assuming both with.
Any help in pointing out if I missed anything would be greatly appreciated.
Thanks,
===============Answer Explanation=====================
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.
3.1. 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.
3.2. 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
3.3. 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
3.4. This means that triangle BCA is an equilateral triangle.
3.5. 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
3.6. Area = .5bh"¨Area = .5(12)(1081/2) = 6*1081/2
3.7. Statement (1) is SUFFICIENT
4. Evaluate Statement (2) alone.
4.1. 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.
4.2. Using the Pythagorean theorem, DG = 1081/2"¨(CG)2 + (DG)2 = (CD)2"¨62 + (DG)2 = 122"¨DG = 1081/2
4.3. 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).
4.4. 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.
4.5. 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.
4.6. 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
4.7. Statement (2) alone is SUFFICIENT.
5. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.
That's a great question: nothing the Platinum GMAT said was wrong, but they were playing fast and loose with geometry there. Let's dissect that a bit. 
