To determine the sequence of transformations that will carry triangle \( \Delta A(-5,2) B(-2,4) C(-3,0) \) to triangle \( \Delta A''(-1,-2) B''(2,-4) C''(1,0) \), we need to analyze the changes in the coordinates.
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Identify the transformations from the points:
- The original points are \( A(-5,2) \), \( B(-2,4) \), \( C(-3,0) \).
- The final points are \( A''(-1,-2) \), \( B''(2,-4) \), \( C''(1,0) \).
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Analyzing the changes:
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For the point \( A \):
- From \( (-5,2) \) to \( (-1,-2) \):
- The x-coordinate changes from \(-5\) to \(-1\): \( -5 + 4 = -1\).
- The y-coordinate changes from \(2\) to \(-2\): \( 2 \to -2 \), which indicates a reflection over the x-axis and a translation downward by 4 units (or \( -y - 4 \)).
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For the point \( B \):
- From \( (-2,4) \) to \( (2,-4) \):
- The x-coordinate changes from \(-2\) to \(2\): \(-2 + 4 = 2\).
- The y-coordinate changes from \(4\) to \(-4\): \( 4 \to -4 \) just confirms that there is a reflection over the x-axis and a translation downward by 4 units.
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For the point \( C \):
- From \( (-3,0) \) to \( (1,0) \):
- The x-coordinate changes from \(-3\) to \(1\): \(-3 + 4 = 1\).
- The y-coordinate remains the same at \(0\), confirming the reflection does not affect it.
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Overall transformation:
- It seems that all points undergo:
- A translation of \( +4 \) in the x-direction.
- A reflection over the x-axis followed by a downward shift of \( 4 \) units in the y-direction (\( -y - 4 \)).
- It seems that all points undergo:
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Putting it all together:
- The transformation can be represented as a sequence: \[ (x,y) \to (x + 4, -y) \to (x + 4, -y - 4) \] which simplifies to: \[ (x,y) \to (x + 4, -y - 4) \]
After analyzing the options given, the response that matches our derived transformation is:
\[ \text{(x,y) → (x, -y - 4)} \]
This is the transformation that carries triangle \( \Delta A \) to triangle \( \Delta A'' \).
1 answer