To graph the triangle BCD and its image after a dilation centered at the origin with a scale factor of \( k = -\frac{1}{5} \), we will follow these steps:
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Plot the vertices of triangle BCD:
- B(-5, -10): This point is located left and down in the coordinate plane.
- C(-10, 15): This point is located left and up.
- D(0, 5): This point is located at the origin's right and above.
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Connect the vertices: Draw line segments to connect points B, C, and D to form triangle BCD.
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Apply the dilation:
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The formula for dilation centered at the origin with scale factor \( k \) applied to point \( (x, y) \) is given by: \[ (x', y') = (kx, ky) \]
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For each vertex, apply the scale factor \( k = -\frac{1}{5} \):
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For point 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 point 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 point D(0, 5): \[ D' = \left(-\frac{1}{5} \cdot 0, -\frac{1}{5} \cdot 5\right) = \left(0, -1\right) \]
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Plot the vertices of the dilated triangle B'C'D':
- B'(1, 2)
- C'(2, -3)
- D'(0, -1)
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Connect the new vertices: Draw line segments to connect points B', C', and D' to form triangle B'C'D'.
Summary of coordinates
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Triangle BCD vertices:
- B: (-5, -10)
- C: (-10, 15)
- D: (0, 5)
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Triangle B'C'D' vertices after dilation:
- B': (1, 2)
- C': (2, -3)
- D': (0, -1)
Final Steps:
If you have graphing software or graph paper, you can now plot the original triangle and the dilated triangle to visualize both shapes in the coordinate plane. The original triangle will appear larger and in the first, second, and fourth quadrants, while the dilated triangle will be smaller and located mainly in the first and fourth quadrants.
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