Question

what is the final transformation in the composition of transformations that maps pre-image abcd to image a\b\c\d\?

a translation down and to the right
a translation up and to the right
a 270° rotation about point b
a 180° rotation about point b

Explanation:

Identify coordinates of the figures

We find the coordinates of the vertices for each shape from the grid:

• Pre-image \(ABCD\):

\(A(5, 7)\), \(B(3.5, 4)\), \(C(9, 4)\), \(D(7, 7)\)

• Intermediate image \(A'B'C'D'\):

\(A'(-6, -2.5)\), \(B'(-7.5, -6)\), \(C'(-2, -6)\), \(D'(-4, -2.5)\)

• Final image \(A''B''C''D''\):

\(A''(-5, -7.5)\), \(B''(-7.5, -6)\) (noting \(B''\) is at the same position as \(B'\)), \(C''(-7.5, -11)\), \(D''(-5, -9.5)\)

Analyze the first transformation

We compare the coordinates of \(ABCD\) and \(A'B'C'D'\):

• The horizontal shift is from \(B(3.5, 4)\) to \(B'(-7.5, -6)\), which is a translation of \(-11\) units horizontally and \(-10\) units vertically.
• Checking other vertices:

\(A(5, 7)
ightarrow (-6, -3)\) (the grid shows \(A'\) is at \((-6, -3)\) and \(D'\) is at \((-4, -3)\)).

• This confirms the first transformation is a translation mapping \(ABCD\) to \(A'B'C'D'\).

Analyze the final transformation

We examine the mapping from \(A'B'C'D'\) to \(A''B''C''D''\):

• Point \(B'\) is at \((-7.5, -6)\), and \(B''\) is also at \((-7.5, -6)\). Since \(B'\) remains fixed, the transformation is a rotation centered at \(B'\).
• Let us find the rotation angle by comparing the orientation of the segments:
• Vector \(\vec{B'C'} = (-2 - (-7.5), -6 - (-6)) = (5.5, 0)\) (pointing horizontally to the right).
• Vector \(\vec{B'C''} = (-7.5 - (-7.5), -11 - (-6)) = (0, -5)\) (pointing vertically downwards).
• Rotating a horizontal vector pointing right \((5.5, 0)\) to a vertical vector pointing down \((0, -5.5)\) requires a \(270^\circ\) counterclockwise rotation (or a \(90^\circ\) clockwise rotation) about the center \(B'\).
• Therefore, the final transformation is a \(270^\circ\) rotation about point \(B'\).

Answer:

• a translation down and to the right
• a translation up and to the right
• a 270° rotation about point B' (Correct answer)
• a 180° rotation about point B'