Question

which set of points represents the image of $\triangle qrs$ after a translation along $\langle -12, 9 \
angle$ followed by a reflection in the line $y = 4$?\
\
(image of a coordinate grid with points: $s(13, -4)$, $q(7, -3)$ and another point partially visible)

Explanation:

Step1: Identify original points

From the graph, we have:

• \( Q(7, -3) \)
• \( R(10, -8) \) (assuming the third point \( R \) is at \( (10, -8) \) from the grid, though not fully labeled, but let's proceed with given \( Q \) and \( S \) and infer \( R \) or use \( Q \) and \( S \) for demonstration)
• \( S(13, -4) \)

Step2: Apply translation \( \langle -12, 9

angle \)
The translation rule is \( (x, y) \to (x - 12, y + 9) \)

• For \( Q(7, -3) \): \( (7 - 12, -3 + 9) = (-5, 6) \)
• For \( S(13, -4) \): \( (13 - 12, -4 + 9) = (1, 5) \)
• For \( R(10, -8) \): \( (10 - 12, -8 + 9) = (-2, 1) \)

Step3: Reflect over \( y = 4 \)

The reflection rule over \( y = k \) is \( (x, y) \to (x, 2k - y) \). Here \( k = 4 \), so \( (x, y) \to (x, 8 - y) \)

• For translated \( Q(-5, 6) \): \( (-5, 8 - 6) = (-5, 2) \)
• For translated \( S(1, 5) \): \( (1, 8 - 5) = (1, 3) \)
• For translated \( R(-2, 1) \): \( (-2, 8 - 1) = (-2, 7) \)

(Note: Since the full triangle's third point \( R \) is partially visible, assuming its coordinates for accuracy, but the process is translation then reflection over \( y = 4 \) using the formula \( 2k - y \) where \( k = 4 \))

Answer:

The image points after translation and reflection would be \( Q'(-5, 2) \), \( R'(-2, 7) \), \( S'(1, 3) \) (exact coordinates depend on precise original \( R \), but the method is translation \( \langle -12,9
angle \) then reflection over \( y = 4 \) using \( (x, 8 - y) \) after translation).