guys it's still confusing, we need one of the experts to explain ...

abcdefg wrote:I've attached a picture of the solution. What I don't get is what they mean by the triangles are similar since their corresponding interior angles have the same measure. How do we know that? Thanks.

These triangles have been constructed in a manner that they become similar,as in the lines dropped are perpendicular,so that interior angles become equal and the triangles are similar.
After that Basic Proportionality theorem has been applied to arrive at the solution.

Sorry to bring back an old thread but Ive tried looking this up and I am just not understanding. I guess my first question is where are we getting point F from?

Secondly, When i draw an equilateral triangle to answer the question, I place point X which turns Triangle ABX into an 30-60-90 triangle. Can I just take answer A (The area of triangular region ABX is 32) and multiply by 2 to arrive at the total area of the triangle? (64)

Thank you

ska7945 wrote:In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of side BC. If point R is the midpoint of line segment XC and if point S is the midpoint of line segment YC, what is the area of triangular region RCS ?
(1) The area of triangular region ABX is 32.
(2) The length of one of the altitudes of triangle ABC is 8.

Triangles RCS and ABC:
Side RC = 1/4(AC).
Side SC = 1/4(BC).
The two triangles share angle BCA.

Triangles with a shared angle (BCA) formed by corresponding sides in the same proportion (RC:AC = 1:4, SC:BC = 1:4) are SIMILAR.
Thus, triangle RCS is similar to triangle ABC.
In similar triangles, corresponding bases and heights are in the same proportion as corresponding sides.
Thus, the base of triangle RCS is 1/4 the base of triangle ABC, and the height of triangle RCS is 1/4 the height of triangle ABC.

Area of triangle ABC = (1/2)bh.
Area of triangle RCS = (1/2)*1/4(b)*1/4(h) = (1/16)(1/2)bh.
Thus, the area of triangle RCS is 1/16 the area of triangle ABC.

Question rephrased: What is the area of triangle ABC?

Statement 1: ABX = 32.

In triangle ABX, AX = 1/2(AC).
In other words, the base of triangle ABX is 1/2 the base of triangle ABC.
Triangles ABX and ABC share height BZ.
Since AX = 1/2(AC), and the two triangles have the same height, ABX = 1/2(ABC).
Thus, the area of triangle ABC = 64, and the area of triangle RCS = (1/16)(64) = 4.
SUFFICIENT.

Statement 2: height = 8.
No way to determine the area of triangle ABC or of triangle RCS.
INSUFFICIENT.

The correct answer is A.

Very good question, but if the diagram in the question is not given, then the question is impossible to solve...

Thank you gmatguru for your really helpful explanation...