Question

If X and Y are points in a plane and X lies inside the circle C with center O and radius 2, does Y lie inside circle C?
(1) The length of line segment XY is 3.
(2) The length of line segment OY is 1.5.

• 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

Hint:

When dealing with geometric problems, always consider the relative positions of the points and the circle to determine if sufficient information is provided.

Answer 1

Answer: (E)

Step 1: Analyze statement (1).
Statement (1) tells us that the length of segment \( XY \) is 3. This gives us information about the distance between \( X \) and \( Y \), but it does not give any information about the position of \( Y \) relative to circle \( C \). Specifically, we don't know if \( Y \) lies inside or outside the circle, so statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that the length of segment \( OY \) is 1.5. This means that point \( Y \) is inside the circle because the radius of the circle is 2, and \( OY = 1.5<2 \). However, this statement alone does not tell us anything about the position of point \( X \), so we cannot conclude if \( Y \) lies inside the circle just based on this information. Thus, statement (2) alone is not sufficient.
Step 3: Combine both statements.
Even when combining both statements, we still cannot determine if \( Y \) lies inside the circle. While we know that \( Y \) is within the circle because \( OY = 1.5 \), we do not have enough information about point \( X \) or its relationship with \( Y \). Thus, the combined statements do not provide sufficient information.
\[ \boxed{E} \]