What are the coordinates of the reflection of the point (5, 3) in the horizontal line y = -2?

Introduction / Context:
Reflection of points in lines is an important concept in coordinate geometry and symmetry. When a point is reflected across a horizontal line of the form y = constant, its x coordinate remains unchanged, while its y coordinate is adjusted so that the point lies the same vertical distance on the other side of the line. This question checks whether you can correctly compute that new y coordinate using a simple distance idea.

Given Data / Assumptions:

• Original point P has coordinates (5, 3).
• We reflect P in the horizontal line y = -2.
• The reflection produces a new point P' with the same perpendicular distance from the line on the opposite side.

Concept / Approach:
For reflection across the line y = k, the x coordinate does not change. The y coordinate moves so that the distance above the line is equal to the distance below the line. If the original y value is y1, the new y value y2 satisfies:
distance above = y1 - k distance below = k - y2 To reflect, we set these distances equal: y1 - k = k - y2, and then solve for y2.

Step-by-Step Solution:
Step 1: Identify k in the line y = -2, so k = -2. Step 2: The original point is P(5, 3), so x1 = 5 and y1 = 3. Step 3: Compute the vertical distance from P to the line: y1 - k = 3 - (-2) = 3 + 2 = 5. Step 4: The reflected point P' will be 5 units below the line y = -2. Step 5: So new y coordinate y2 = -2 - 5 = -7. Step 6: The x coordinate remains unchanged at 5. Step 7: Therefore the reflected point is (5, -7).

Verification / Alternative check:
We can verify by checking that the midpoint of the segment joining (5, 3) and (5, -7) lies on the line y = -2. The midpoint y coordinate is (3 + (-7)) / 2 = -4 / 2 = -2. The midpoint is therefore (5, -2), which lies exactly on y = -2, confirming that the reflection has been done correctly.

Why Other Options Are Wrong:

• (-9, 3) and (-9, -3): These options change the x coordinate, which is not correct for reflection in a horizontal line.
• (-5, -7): This changes both x and y and is unrelated to the correct symmetry.
• (5, 7): This lies above the line instead of below and does not maintain the correct distance from y = -2.

Common Pitfalls:
Students sometimes reflect across the x axis by mistake or forget that only the y coordinate changes for reflection in a horizontal line. Another frequent error is subtracting the distances incorrectly when dealing with negative y values. Carefully computing the vertical distance and then moving the same amount to the other side avoids these mistakes.

Final Answer:
Hence, the reflection of the point (5, 3) in the line y = -2 is (5, -7).