In triangle ABC, point X is the midpoint of side AC and point Y is the midpoint of side BC....

Understanding the Question

We have triangle ABC with several midpoints creating smaller regions. Let's trace the relationships:

• X is the midpoint of AC, Y is the midpoint of BC
• R is the midpoint of XC (making R the \(\frac{3}{4}\) point of AC from A, or \(\frac{1}{4}\) point from C)
• S is the midpoint of YC (making S the \(\frac{3}{4}\) point of BC from B, or \(\frac{1}{4}\) point from C)

We need to determine: The area of triangle RCS

What "sufficient" means here: We can calculate the exact numerical value of the area of triangle RCS.

Key Insight

This is a classic geometry problem about area relationships through midpoints. When we connect midpoints in triangles, we create smaller triangles with predictable area ratios. Since R and S are "midpoints of midpoints," they're actually at the quarter-points of their respective sides from C. This creates a specific fractional relationship with the original triangle's area.

Analyzing Statement 1

Statement 1: The area of triangular region ABX is 32.

What Statement 1 Tells Us

Triangle ABX connects vertex B with the midpoint X of side AC. Here's the crucial relationship: when you connect a vertex to the midpoint of the opposite side, you create a triangle with exactly half the area of the original triangle.

Finding the Area of Triangle RCS

Since X is the midpoint of AC:

• Triangle ABX has half the area of triangle ABC
• Given: \(\mathrm{Area(ABX)} = 32\)
• Therefore: \(\mathrm{Area(ABC)} = 64\)

Now for triangle RCS:

• R is at the \(\frac{1}{4}\) point of AC from C
• S is at the \(\frac{1}{4}\) point of BC from C
• When both vertices are at quarter-points of their respective sides from a common vertex, the resulting triangle has area = \(\frac{1}{16}\) × Area of original triangle

Why \(\frac{1}{16}\)? Because:

• The base CR is \(\frac{1}{4}\) of the base CA
• The height from S to line CR is \(\frac{1}{4}\) of the corresponding height in triangle ABC
• Area factor = \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\)

Therefore: \(\mathrm{Area(RCS)} = \frac{64}{16} = 4\)

[STOP - Sufficient!] We found the exact area.

Statement 1 is sufficient.

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The length of one of the altitudes of triangle ABC is 8.

What Statement 2 Provides

We know one altitude of triangle ABC equals 8, but we don't know:

• Which altitude it is (from which vertex?)
• The length of the corresponding base
• The overall shape or size of the triangle

Testing Different Scenarios

Since we don't know which base corresponds to the altitude of 8, let's test two different triangles:

Scenario 1: Altitude = 8, base = 10

• \(\mathrm{Area(ABC)} = \frac{1}{2} \times 10 \times 8 = 40\)
• \(\mathrm{Area(RCS)} = \frac{40}{16} = 2.5\)

Scenario 2: Altitude = 8, base = 20

• \(\mathrm{Area(ABC)} = \frac{1}{2} \times 20 \times 8 = 80\)
• \(\mathrm{Area(RCS)} = \frac{80}{16} = 5\)

Since different triangle configurations lead to different areas for RCS, we cannot determine a unique value.

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Statement 1 alone gives us the exact area of triangle RCS through geometric relationships, while Statement 2 leaves multiple possibilities open.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."