What are the coordinates of the reflection of the point (-0.5, 6) in the x-axis?

Introduction / Context:
Reflections in coordinate geometry are a basic but important concept, often used to describe symmetry. Reflecting a point across the x-axis changes only the vertical position of the point, while its horizontal position remains the same. This question tests whether you understand how the y coordinate changes while the x coordinate is preserved during a reflection in the x-axis.

Given Data / Assumptions:

• The original point has coordinates (-0.5, 6).
• We reflect this point in the x-axis.
• The x-axis is the horizontal line defined by y = 0.
• We must find the coordinates of the resulting reflected point.

Concept / Approach:
When a point (x, y) is reflected across the x-axis, its x coordinate remains unchanged and its y coordinate changes its sign. In other words, a point (x, y) becomes (x, -y). This rule follows from the property that the x-axis is the set of points with y = 0, and reflection across it reverses the vertical distance above or below this axis.

Step-by-Step Solution:
Step 1: Start with the original point P having coordinates (-0.5, 6). Step 2: Identify the x coordinate x = -0.5 and the y coordinate y = 6. Step 3: Reflect across the x-axis by keeping x the same and changing y to -y. Step 4: Compute the new y coordinate: -y = -6. Step 5: The reflected point P' therefore has coordinates (-0.5, -6). Step 6: This point lies directly below the original point at the same horizontal position.

Verification / Alternative check:
The distance from the original point to the x-axis is 6 units (since its y coordinate is 6). After reflection, the new point should be 6 units below the x-axis, so its y coordinate should be -6. The midpoint of the segment joining (-0.5, 6) and (-0.5, -6) is (-0.5, 0), which lies on the x-axis. This confirms that the reflection has been done correctly.

Why Other Options Are Wrong:

• (0.5, -6): This changes the sign of both coordinates, not just y, so it is a reflection across both axes.
• (-6, -0.5) and (6, -0.5): These options swap roles of x and y and do not reflect correctly across the x-axis.
• (-0.5, 6): This is the original point without any reflection applied.

Common Pitfalls:
Students sometimes confuse reflections across the x-axis and y-axis, or mistakenly change both coordinates. Always remember: reflection in the x-axis changes only the sign of the y coordinate. Reflection in the y-axis changes only the sign of the x coordinate. Keeping this distinction clear helps avoid confusion in coordinate geometry questions.

Final Answer:
Therefore, the reflection of the point (-0.5, 6) in the x-axis is (-0.5, -6).