Question

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.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is A Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

• As shown in the figure, X and Y are the midpoints of AC and BC, respectively, of , and R and S are the midpoints of XC and YC, respectively. Thus, letting AC = b it follows that and . Also, if RF, YG, and SH are perpendicular to AC as shown, then , and are similar triangles, since their corresponding interior angles have the same measure. Thus, letting BF = h, it follows that and . The area of , which is , can be determined exactly when the value of bh can be determined.
  

1. Given that the area of , which is , or , is 32, then bh = 4 * 32; SUFFICIENT.
2. Without knowing the length of the side to which the altitude is drawn, the area of , and hence the value of bh, cannot be determined; NOT sufficient. The correct answer is A; statement 1 alone is sufficient.