Geometry

Question Analysis:
This problem requires reversing a glide reflection. You are given the final image, the reflection line, and the translation vector, and you need to determine the original point.
The main focus is on understanding how to reverse both translation and reflection in proper order.

Key Concept Explanation:
A glide reflection consists of a reflection over a line followed by a translation parallel to that line.
To reverse this:
Undo the translation (subtract the translation vector).
Then undo the reflection (reflect back over the same mirror line).

Step-by-Step Solution:
Given data:
Final point P′ = (9, 2)
Reflection line: y = 4
Translation vector: (3, 0)
Step 1: Undo the translation.
The translation vector is (3, 0), so to undo the translation, subtract the vector from P′:
P = (9 – 3, 2 – 0) = (6, 2)
Step 2: Undo the reflection.
The point P′(6, 2) is reflected across the line y = 4. To reverse the reflection, find the point that, when reflected across y = 4, gives the point P′(6, 2).
The distance between y = 4 and y = 2 is 2 units.
Since the reflected point is below y = 4, the original point must be 2 units above y = 4.
Therefore, the original point P has a y-coordinate of 6.
So, the original point P is (6, 6).

Option Analysis:...