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.
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