Is the point Q on the circle with center C ? R is a point on the circle and the...

Understanding the Question

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).

What We Need to Determine

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:

• Either know the exact relationship between QC and the radius
• Or have enough geometric constraints to uniquely determine Q's position

Analyzing Statement 1

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:

• R is on the circle → \(\mathrm{CR} = \mathrm{radius}\)
• \(\mathrm{QR} = \mathrm{QC}\)

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.

Testing Different Scenarios

Let me visualize specific cases to show why this isn't sufficient:

Case 1: Q inside the circle

• Imagine the radius is 5 units and Q is at distance 3 from C
• Since Q is closer to C than to the circle's edge, we can find a point R on the circle where QR = 3
• This satisfies \(\mathrm{QR} = \mathrm{QC} = 3\) ✓

Case 2: Q on the circle

• Now place Q on the circle itself, so \(\mathrm{QC} = 5\) (the radius)
• We can find a different point R on the circle where \(\mathrm{QR} = 5\)
• This satisfies \(\mathrm{QR} = \mathrm{QC} = 5\) ✓

Case 3: Q outside the circle

• Finally, place Q outside at distance 7 from C
• We can still find a point R on the circle where \(\mathrm{QR} = 7\)
• This satisfies \(\mathrm{QR} = \mathrm{QC} = 7\) ✓

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.

Analyzing Statement 2

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:

• \(\mathrm{SC} = \mathrm{radius}\)
• \(\mathrm{QS} = \mathrm{SC} = \mathrm{radius}\)

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.

Testing Different Scenarios

Case 1: Q at the center C

• If Q coincides with C, then \(\mathrm{QS} = \mathrm{CS} = \mathrm{radius}\) ✓
• Here, Q is inside the circle (at the center)

Case 2: Q on the circle

• There are exactly two positions where Q can be on the original circle and \(\mathrm{QS} = \mathrm{radius}\)
• These occur when triangle QSC forms an equilateral triangle (all sides equal to the radius)
• This creates \(60°\) angles, and Q is on the circle ✓

Case 3: Q outside the circle

• Q can also be positioned outside the circle where QS still equals the radius
• For example, if Q is on the extension of line CS beyond C
• Here Q is outside but \(\mathrm{QS} = \mathrm{radius}\) ✓

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.

Combining Statements

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:

• \(\mathrm{QR} = \mathrm{QC}\) (Statement 1)
• \(\mathrm{QS} = \mathrm{SC} = \mathrm{radius}\) (Statement 2)
• Both R and S are on the circle

Even with both geometric constraints, Q's position is still not uniquely determined. Here's why:

The Key Insight

Think of it this way:

• Statement 1 places Q somewhere on a certain locus of points (where \(\mathrm{QR} = \mathrm{QC}\))
• Statement 2 places Q somewhere on another locus of points (where \(\mathrm{QS} = \mathrm{radius}\))
• These two loci can intersect at multiple points

We can construct configurations where:

1. Q is inside the circle and both conditions are satisfied
2. Q is on the circle and both conditions are satisfied
3. Q is outside the circle and both conditions are satisfied

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!]

The Answer: E

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.