Question

in the similarity transformation of \\(\delta abc\\) to \\(\delta fed\\), \\(\delta abc\\) was dilated by a scale factor of \\(\frac{1}{2}\\), reflected across the ? , and moved through the translation .

origin
\\(y = 2\\)
\\(y\\)-axis
\\(x\\)-axis

Explanation:

Identify original coordinates

Find coordinates of \(\Delta ABC\):
\(A = (-8, 2)\)
\(B = (-4, 4)\)
\(C = (-2, 2)\)

Apply dilation transformation

Using the Coordinate Dilation knowledge point:
Scale factor is \(\frac{1}{2}\).
\(A' = (-4, 1)\)
\(B' = (-2, 2)\)
\(C' = (-1, 1)\)

Analyze target coordinates

Find coordinates of \(\Delta FED\):
\(F = (2, 1)\)
\(E = (3, 1.5)\)
\(D = (5, 1)\)

Determine reflection axis

Using the Coordinate Reflection knowledge point:
Compare orientation of \(\Delta A'B'C'\) and \(\Delta FED\).
\(A'\) is leftmost, \(C'\) is rightmost, \(B'\) is top.
\(F\) is leftmost, \(D\) is rightmost, \(E\) is top.
No vertical reflection occurred.
To map negative \(x\)-coordinates to positive, reflect across the \(y\)-axis.
Using the Similarity Transformations knowledge point:
Reflecting \(\Delta A'B'C'\) across the \(y\)-axis:
\(A'' = (4, 1)\)
\(B'' = (2, 2)\)
\(C'' = (1, 1)\)

Determine translation vector

Using the Coordinate Translation knowledge point:
Translate \(\Delta A''B''C''\) to \(\Delta FED\):
\(F - C'' = (2, 1) - (1, 1) = (1, 0)\)
\(E - B'' = (3, 1.5) - (2, 2) = (1, -0.5)\)
\(D - A'' = (5, 1) - (4, 1) = (1, 0)\)
Thus, the reflection is across the \(y\)-axis.

Answer:

• origin
• y = 2
• y-axis (Correct answer)
• x-axis