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Is the point Q on the circle with center C ?
Let's understand what we're being asked: Is the point Q on the circle with center C?
This is a yes/no question. We need to determine whether we can definitively say YES (Q is on the circle) or NO (Q is not on the circle).
For a point to be on a circle, its distance from the center must equal the radius. So Q is on the circle if and only if \(\mathrm{QC} = \mathrm{radius}\).
To answer this question with certainty, we need information that allows us to:
Statement 1: R is a point on the circle and the distance from Q to R equals the distance from Q to C.
Let's translate this:
This creates an isosceles triangle QRC where \(\mathrm{QR} = \mathrm{QC}\). But here's the key insight: this condition can be satisfied with Q at many different positions relative to the circle.
Let me visualize specific cases to show why this isn't sufficient:
Case 1: Q inside the circle
Case 2: Q on the circle
Case 3: Q outside the circle
Since Q can be inside, on, or outside the circle while satisfying Statement 1, we get different answers (NO, YES, NO) to our question.
Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: S is a point on the circle and the distance from Q to S equals the distance from S to C.
Since S is on the circle:
This means Q lies on a circle centered at S with radius equal to the original circle's radius. Let's see what positions are possible for Q.
Case 1: Q at the center C
Case 2: Q on the circle
Case 3: Q outside the circle
Since Q can be inside (at center), on, or outside the circle while satisfying Statement 2, we again get different answers to our question.
Statement 2 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
Since we've eliminated A, B, and D, we need to check if both statements together are sufficient (choice C) or not sufficient (choice E).
From both statements together:
Even with both geometric constraints, Q's position is still not uniquely determined. Here's why:
Think of it this way:
We can construct configurations where:
The two constraints don't provide enough information to pin down Q's exact location. Multiple positions for Q can satisfy both conditions simultaneously.
Therefore, even with both statements, we cannot definitively answer whether Q is on the circle.
The statements together are NOT sufficient.
[STOP - Not Sufficient!]
Since Q could be inside, on, or outside the circle while satisfying both conditions, we cannot determine with certainty whether Q is on the circle.
Answer Choice E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.