Point P is the midpoint of segment AB. The coordinates of point P are (2, 1) and the coordinates of point A are (11, 5). What are the coordinates of point B?

Introduction / Context:
This is a straightforward coordinate geometry question that uses the midpoint formula for a line segment in the Cartesian plane. The problem asks you to find the coordinates of an unknown endpoint of a segment when the midpoint and the other endpoint are given. Such questions test your ability to manipulate basic formulas and to work with simple algebraic equations involving averages of coordinates.

Given Data / Assumptions:

• Point P is the midpoint of segment AB.
• Coordinates of P are (2, 1).
• Coordinates of A are (11, 5).
• Coordinates of B are unknown and must be found.
• We work on the standard x-y coordinate plane using the usual midpoint formula.

Concept / Approach:
For a segment joining points A(x1, y1) and B(x2, y2), the coordinates of the midpoint P are given by:
P_x = (x1 + x2) / 2 P_y = (y1 + y2) / 2 We are given P and A, and we need to reverse this relationship to solve for the unknown coordinates of B. This means we will set up equations based on these midpoint formulas and solve for x2 and y2.

Step-by-Step Solution:
Step 1: Let B have coordinates (x, y). Step 2: Use the midpoint formula for the x-coordinate: (11 + x) / 2 = 2. Step 3: Multiply both sides by 2: 11 + x = 4, so x = 4 - 11 = -7. Step 4: Use the midpoint formula for the y-coordinate: (5 + y) / 2 = 1. Step 5: Multiply both sides by 2: 5 + y = 2, so y = 2 - 5 = -3. Step 6: Therefore B = (-7, -3). Step 7: Compare this with the options, and we see that option (-7, -3) exactly matches our result.

Verification / Alternative check:
To verify, substitute A and B back into the midpoint formula. For x-coordinates: (11 + (-7)) / 2 = 4 / 2 = 2, which matches the given midpoint x-coordinate. For y-coordinates: (5 + (-3)) / 2 = 2 / 2 = 1, which matches the given midpoint y-coordinate as well. Both coordinates of the midpoint check out, confirming that B is indeed (-7, -3).

Why Other Options Are Wrong:
The point (6.5, 3) gives midpoint x-coordinate (11 + 6.5) / 2 = 8.75 which is not equal to 2. The point (7, 3) gives midpoint x-coordinate (11 + 7) / 2 = 9 which is incorrect. The point (-6.5, -3) does not produce integer midpoint coordinates and fails to give P = (2, 1). The point (9, -1) also produces a different midpoint. Only (-7, -3) satisfies both coordinates of the midpoint formula.

Common Pitfalls:
A common mistake is to forget that the midpoint takes the average of the coordinates, not the difference. Some students also write equations like (x - 11) / 2 instead of (x + 11) / 2. Another frequent error is to mix up the formulas for midpoint and distance, or to solve only one of the two coordinates and then guess the other. Careful use of both x and y midpoint formulas ensures the correct point is found.

Final Answer:
Thus, the coordinates of point B are (-7, -3).