Point P with coordinates (8, 5) is the midpoint of line segment AB. The coordinates of A are (5, y) and the coordinates of B are (x, -3). What is the value of x, the x coordinate of point B?

Introduction / Context: This question checks your understanding of the midpoint formula in coordinate geometry. Midpoints are very common in analytic geometry problems because they relate the endpoints of a segment to the point exactly in the middle. Knowing how to handle x and y coordinates separately is essential for solving such questions quickly and accurately.

Given Data / Assumptions: - Midpoint P has coordinates (8, 5). - Endpoint A has coordinates (5, y). - Endpoint B has coordinates (x, -3). - The point P lies exactly halfway between A and B on the segment AB. - We need to find x, the x coordinate of B. The y coordinate y of A is not asked in this question.

Concept / Approach: If a point P(xm, ym) is the midpoint of a segment joining A(x1, y1) and B(x2, y2), the midpoint formula is: xm = (x1 + x2) / 2 ym = (y1 + y2) / 2 In other words, the midpoint coordinates are simply the averages of the x coordinates and the y coordinates of the endpoints. Since we know P, A and the y coordinate of B, we can set up equations and solve for x and y independently.

Step-by-Step Solution: Step 1: Apply the midpoint formula for the x coordinates: 8 = (5 + x) / 2. Step 2: Multiply both sides by 2 to remove the denominator: 16 = 5 + x. Step 3: Solve for x by subtracting 5 from both sides: x = 16 - 5 = 11. Step 4: For completeness, you can apply the midpoint formula for the y coordinates: 5 = (y + (-3)) / 2. Step 5: From this, y + (-3) = 10, which gives y = 13, but this value is not required for the final answer.

Verification / Alternative check: Check the x coordinate: with A(5, 13) and B(11, -3), the average of the x coordinates is (5 + 11) / 2 = 16 / 2 = 8, which matches the x coordinate of P. For the y coordinates, (13 + (-3)) / 2 = 10 / 2 = 5, which matches the y coordinate of P. This confirms that P really is the midpoint and that x = 11 is consistent.

Why Other Options Are Wrong: -11 and -7 would produce a midpoint x coordinate that is negative or too small and would not match xm = 8. 7 also gives (5 + 7) / 2 = 6, not 8, so it is incorrect. 5 would give (5 + 5) / 2 = 5, again not equal to 8.

Common Pitfalls: A common mistake is to confuse midpoint with distance and try to use distance formulas instead of simple averages. Others forget to multiply by 2 or mishandle the algebraic signs when solving for x. Always treat x and y coordinates independently and carefully apply the midpoint formula component wise.

Final Answer: The value of x, the x coordinate of point B, is 11.