To graph the vertices of quadrilateral ABCD and its dilated image centered at the origin with a scale factor of \( k = -\frac{1}{5} \), we first need to identify the coordinates of vertex A. Since the coordinates for points B, C, and D are provided, we will assume a hypothetical coordinate for point A to create a complete quadrilateral.
Given Vertices:
- B(-5, -10)
- C(-10, 15)
- D(0, 5)
For the sake of simplicity, let's assume point A is at (0, 0). You can use any other values for point A as needed.
Step 1: Plotting the Original Quadrilateral ABCD
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Plot the points on a graph:
- Point A(0, 0)
- Point B(-5, -10)
- Point C(-10, 15)
- Point D(0, 5)
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Connect the points to form quadrilateral ABCD:
- Connect A to B
- Connect B to C
- Connect C to D
- Connect D to A
Step 2: Dilating the Points
To find the new coordinates after the dilation with a scale factor of \( k = -\frac{1}{5} \), use the formula: \[ (x', y') = (k \cdot x, k \cdot y) \]
Calculate the new coordinates for each point:
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Point A(0, 0): \[ A' = \left(-\frac{1}{5} \cdot 0, -\frac{1}{5} \cdot 0\right) = (0, 0) \]
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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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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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Point D(0, 5): \[ D' = \left(-\frac{1}{5} \cdot 0, -\frac{1}{5} \cdot 5\right) = \left(0, -1\right) \]
Step 3: Plotting the Dilated Quadrilateral A'B'C'D'
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Plot the new points:
- A'(0, 0)
- B'(1, 2)
- C'(2, -3)
- D'(0, -1)
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Connect the points to form quadrilateral A'B'C'D':
- Connect A' to B'
- Connect B' to C'
- Connect C' to D'
- Connect D' to A'
Final Graph
You should end up with two quadrilaterals:
- The original quadrilateral ABCD with vertices at B(-5, -10), C(-10, 15), D(0, 5), and A(0, 0).
- The dilated quadrilateral A'B'C'D' with coordinates A'(0, 0), B'(1, 2), C'(2, -3), and D'(0, -1).
These steps will yield a graphical representation of both quadrilaterals. If you use graph paper or a graphing software, you can plot these points accordingly for the best visual representation.
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