To graph triangle BCD and its dilated image after applying the dilation centered at the origin with a scale factor of \( k = -\frac{1}{5} \), we can follow these steps:
Step 1: Plot the Original Triangle BCD
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Plot the vertices:
- B (-5, -10): Left 5 units and down 10 units.
- C (-10, 15): Left 10 units and up 15 units.
- D (0, 5): No movement along the x-axis and up 5 units.
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Draw the triangle: Connect the points B, C, and D to form triangle BCD.
Step 2: Perform the Dilation
The formula for dilation centered at the origin with a scale factor \( k \) is: \[ (x', y') = (k \cdot x, k \cdot y) \]
For the scale factor \( k = -\frac{1}{5} \), we will apply this to each vertex.
Calculate the new coordinates:
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For vertex B (-5, -10): \[ B' = \left(-\frac{1}{5} \cdot -5, -\frac{1}{5} \cdot -10\right) = \left(1, 2\right) \]
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For vertex C (-10, 15): \[ C' = \left(-\frac{1}{5} \cdot -10, -\frac{1}{5} \cdot 15\right) = \left(2, -3\right) \]
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For vertex D (0, 5): \[ D' = \left(-\frac{1}{5} \cdot 0, -\frac{1}{5} \cdot 5\right) = (0, -1) \]
Step 3: Plot the Dilated Image B'C'D'
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Plot the new vertices:
- B' (1, 2): Right 1 unit and up 2 units.
- C' (2, -3): Right 2 units and down 3 units.
- D' (0, -1): No movement along the x-axis and down 1 unit.
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Draw the new triangle: Connect the points B', C', and D' to form triangle B'C'D'.
Final Step: Graph Both Triangles
Ensure both triangles BCD and B'C'D' are visible. The original triangle BCD should be in one color (e.g., blue) and its image B'C'D' in a different color (e.g., red).
This illustrates how the points have been dilated and provides a clear comparison between the original and the transformed shapes.