Geometry

Question Analysis:
This question involves reversing a sequence of geometric transformations, which is a higher-order geometry skill.
The main focus is on understanding how to undo a translation and a reflection across y = x.

Key Concept Explanation:
Transformations were applied in this order:
Reflection across y = x → (x, y) → (y, x)
Translation by (+2, –1) → add 2 to x, subtract 1 from y
To find the original point, we reverse the operations in the reverse order:
Undo the translation: subtract the vector
Undo the reflection: swap the coordinates

Step-by-Step Solution:
We’re given:
Final image: P′ = (4, –7)
Step 1 — Undo the translation:
Since the point was moved +2 in x and –1 in y, reverse it:
(4 – 2, –7 + 1) = (2, –6)
Step 2 — Undo the reflection over y = x:
Reflection over y = x: (a, b) → (b, a)
So undo by swapping: (2, –6) →...