Geometry

Question Analysis:
This question tests understanding of sequential transformations and also requires an analysis of the effect of reflection on orientation.
The main focus is on applying reflection + translation, and then making geometric conclusions about location, congruence, and orientation.

Key Concept Explanation:
Reflection over a vertical line x = c: x′ = 2c − x, y′ = y
Translation:
Shift x by h, y by k → (x + h, y + k)
Orientation:
A reflection always reverses orientation (e.g., clockwise becomes counterclockwise).
A translation preserves orientation.
→ So, overall, this transformation reverses the triangle’s orientation.

Step-by-Step Solution:
Step 1: Reflect triangle ABC over x = 2
Original points:
A(1, 2) → reflected: A′ = (2 × 2 − 1, 2) = (3, 2)
B(3, 5) → A′ = (1, 5)
C(4, 2) → A′ = (0, 2)
So triangle A′B′C′ = A′(3, 2), B′(1, 5), C′(0, 2)
Step 2: Translate each point 4 units right and 1 unit down → (x + 4, y − 1)
A′(3, 2) → A″ = (7, 1)
B′(1, 5) → B″ = (5, 4)
C′(0, 2) → C″ = (4, 1)
Now the final triangle A″B″C″ = (7, 1), (5, 4), (4, 1)

Option Analysis:
A. ✅ Correct
Calculated A″ = (7, 1)
Reflection causes orientation reversal
Translation does not affect congruence → triangle is...