Question

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 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 a

Explanation:

Step1: Analyze each transformation option

First, we need to recall the properties of reflections and rotations. A reflection over \( y = -2 \) will flip the figure over the horizontal line \( y = -2 \). A \( 90^\circ \) clockwise rotation about a point \( (x_0,y_0) \) has the transformation rule: for a point \( (x,y) \), the new coordinates \( (x',y') \) are given by \( (x - x_0,y - y_0) \), then rotate \( 90^\circ \) clockwise: \( (y - y_0,-(x - x_0)) \), then add back \( (x_0,y_0) \) to get \( (y - y_0 + x_0,-(x - x_0)+y_0) \).

Let's consider the positions of the figures. The blue triangle has a vertex \( A \) and the other figure (the V - shape) has a vertex \( B \).

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

First, reflection over \( y = -2 \): For a point \( (x,y) \), the reflected point over \( y=-2 \) is \( (x,-4 - y) \) (since the distance from \( y \) to \( -2 \) is \( |y + 2| \), so the new \( y \) - coordinate is \( -2-|y + 2| \) if \( y>-2 \), or \( -2 + |y + 2| \) if \( y < -2 \); in general, \( y'=-2-(y + 2)=-4 - y \) when \( y>-2 \)).

After reflection, we then rotate \( 90^\circ \) clockwise about \( A \). Let's assume the coordinates of the relevant points. Suppose \( A=(x_A,y_A) \). After reflection, a point \( P=(x,y) \) becomes \( P_1=(x,-4 - y) \). Then, to rotate \( 90^\circ \) clockwise about \( A \), we translate the coordinate system so that \( A \) is at the origin: \( (x - x_A,-4 - y - y_A) \), rotate \( 90^\circ \) clockwise: \( (-4 - y - y_A,-(x - x_A)) \), then translate back: \( (-4 - y - y_A+x_A,-(x - x_A)+y_A) \).

By visually inspecting the figures, the reflection over \( y = -2 \) aligns the triangle in a position such that a subsequent \( 90^\circ \) clockwise rotation about \( A \) will map it to the V - shaped figure.

Let's check other options:

Option A: Reflection over \( y=-2 \) then rotation about \( B \): The center of rotation is \( B \), but the final figure's position does not match when we rotate about \( B \) after reflection.

Option B: Rotation about \( A \) first then reflection over \( y=-2 \): Rotating about \( A \) first will not align the figure to be reflected over \( y=-2 \) to get the V - shape.

Option C: Rotation about \( B \) first then reflection over \( y=-2 \): Rotating about \( B \) first and then reflecting will not give the correct figure.

Answer:

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