Question

a 90° clockwise rotation about point b, followed by a reflection over the line y = -2. a reflection over the line y = -2, followed by a 90° clockwise rotation about point b. a 90° clockwise rotation about point a, followed by a reflection over the line y = -2. a reflection over the line y = -2, followed by a 90° clockwise rotation about point a.

Explanation:

Step1: Analyze Rotation Center

First, identify the center of rotation. For a rotation about a point, the distance from the center to each vertex should be consistent before and after rotation. Looking at the figure, point A or B? Let's check the options. The key is to see which transformation sequence maps the original triangle to the target (the small triangle at the top). Let's consider the rotation center. If we rotate about point A, the distance from A to C and A to B should be preserved. Alternatively, about B. But let's check the reflection over \( y = -2 \).

Step2: Check Transformation Order

Now, check the order: reflection first or rotation first. Let's take an option, say, reflection over \( y = -2 \) followed by 90° clockwise rotation about A. Wait, maybe better to track a vertex. Let's take vertex C. Original position: let's say coordinates (approx). After reflection over \( y = -2 \), then rotation about A. Or rotation about A first, then reflection. Wait, the correct option: let's see the target triangle. The small triangle at the top has a vertex at (3, -1) maybe? Wait, original triangle: A is at (let's say) (2, -3), B at (3, 5), C at (-2, 4). Wait, maybe not. Alternatively, the correct transformation: a reflection over \( y = -2 \) (which is a horizontal line) changes the y-coordinate: \( y' = -2 - (y - (-2)) = -4 - y \). Then a 90° clockwise rotation about point A: the formula for 90° clockwise rotation about (h,k) is \( (x', y') = (h + (y - k), k - (x - h)) \). Wait, maybe the correct option is: A reflection over the line \( y = -2 \), followed by a \( 90^\circ \) clockwise rotation about point A. Wait, no, let's check the options again.

Wait, the options are:

1. 90° clockwise rotation about B, then reflection over \( y = -2 \)
2. Reflection over \( y = -2 \), then 90° clockwise rotation about B
3. 90° clockwise rotation about A, then reflection over \( y = -2 \)
4. Reflection over \( y = -2 \), then 90° clockwise rotation about A

Let's track a vertex. Let's take the bottom vertex (A) and the other vertices. Wait, the original triangle has A at the bottom, B at the top, C at the left. The target triangle is small, above. Let's see: if we first reflect over \( y = -2 \), the bottom vertex A (which is on \( y = -3 \)) would reflect to \( y = -2 - (-3 - (-2)) = -2 - (-1) = -1 \), so A would be at (2, -1) after reflection? Wait, no: reflection over \( y = k \) is \( y' = 2k - y \). So for \( k = -2 \), \( y' = 2(-2) - y = -4 - y \). So if A is at (x, -3), then after reflection, \( y' = -4 - (-3) = -1 \), so A is at (x, -1). Then a 90° clockwise rotation about A: the rotation of a point (x, y) about (h, k) 90° clockwise is \( (h + (y - k), k - (x - h)) \). Let's take another vertex, say C. Original C: let's say (x1, y1). After reflection, (x1, -4 - y1). Then rotate about A (h, k) = (x, -1). So \( x' = x + (-4 - y1 - (-1)) = x - 3 - y1 \), \( y' = -1 - (x1 - x) \). Does this match the target? Alternatively, maybe the correct option is: A reflection over the line \( y = -2 \), followed by a \( 90^\circ \) clockwise rotation about point A. Wait, no, let's check the options again. Wait, the correct answer is the last option: "A reflection over the line \( y = -2 \), followed by a \( 90^\circ \) clockwise rotation about point A". Wait, no, let's see the options:

Options:

1. 90° clockwise rotation about B, then reflection over \( y = -2 \)
2. Reflection over \( y = -2 \), then 90° clockwise rotation about B
3. 90° clockwise rotation about A, then reflection over \( y = -2 \)
4. Reflection over \( y = -2 \), then 90° clockwise rotation about A

Wait, maybe I made a mistake. Let's think about the center of rotation. The small triangle at the top: its position relative to A. If we rotate about A, then reflect, or reflect then rotate about A. Let's take a vertex, say the bottom vertex of the original triangle (A) and the top vertex of the small triangle. After reflection over \( y = -2 \), A moves to (x, -1) (since original y is -3, reflection over \( y = -2 \) gives y = -1). Then a 90° clockwise rotation about A: the rotation would move the other vertices around A. Alternatively, if we rotate about A first, then reflect, the order matters. But the correct option is the fourth one: "A reflection over the line \( y = -2 \), followed by a \( 90^\circ \) clockwise rotation about point A". Wait, no, let's check the answer. Wait, maybe the correct option is D (the fourth one: "A reflection over the line \( y = -2 \), followed by a \( 90^\circ \) clockwise rotation about point A").

Answer:

D. A reflection over the line \( y = -2 \), followed by a \( 90^\circ \) clockwise rotation about point A