Question

triangle \\(abc\\) is rotated about the origin \\(180^\circ\\) clockwise and then translated 6 units to the left. this sequence of transformations produces the image, triangle \\(abc\\), shown on the graph.

the coordinates of vertex \\(a\\) in the preimage triangle \\(abc\\) are \\(a(\quad, \quad)\\).

Explanation:

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<reasoning>

Identify the coordinates of the transformed vertex

From the given graph, the coordinates of the vertex \(A'\) of the image triangle \(A'B'C'\) are:
\[
A' = (-3, -6)
\]

Reverse the translation transformation

The sequence of transformations applied to the preimage \(ABC\) to get \(A'B'C'\) is:

1. Rotate \(180^\circ\) about the origin.
2. Translate \(6\) units to the left.

To find the preimage vertex \(A\), we must apply the inverse transformations in reverse order:

1. Translate \(6\) units to the right:

\[
(-3 + 6, -6) = (3, -6)
\]

Reverse the rotation transformation

1. Rotate \(180^\circ\) about the origin (the inverse of a \(180^\circ\) rotation is also a \(180^\circ\) rotation, which maps \((x, y) \to (-x, -y)\)):

\[
(3, -6) \to (-3, 6)
\]
</reasoning>

<answer>
The coordinates of vertex A in the preimage triangle ABC are A( <blank>-3</blank> , <blank>6</blank> ).
</answer>

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"Mathematics",
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Answer:

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</pre_analysis>

<reasoning>

Identify the coordinates of the transformed vertex

From the given graph, the coordinates of the vertex \(A'\) of the image triangle \(A'B'C'\) are:
\[
A' = (-3, -6)
\]

Reverse the translation transformation

The sequence of transformations applied to the preimage \(ABC\) to get \(A'B'C'\) is:

1. Rotate \(180^\circ\) about the origin.
2. Translate \(6\) units to the left.

To find the preimage vertex \(A\), we must apply the inverse transformations in reverse order:

1. Translate \(6\) units to the right:

\[
(-3 + 6, -6) = (3, -6)
\]

Reverse the rotation transformation

1. Rotate \(180^\circ\) about the origin (the inverse of a \(180^\circ\) rotation is also a \(180^\circ\) rotation, which maps \((x, y) \to (-x, -y)\)):

\[
(3, -6) \to (-3, 6)
\]
</reasoning>

<answer>
The coordinates of vertex A in the preimage triangle ABC are A( <blank>-3</blank> , <blank>6</blank> ).
</answer>

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