To graph quadrilateral JKLM and its image after a dilation centered at the origin with a scale factor of \( k = 0.25 \), follow these steps:
Step 1: Plot the Original Vertices
We start by plotting the original vertices of the quadrilateral JKLM:
- J: \( (1, 0) \)
- K: \( (-8, 4) \)
- L: \( (0, -4) \)
- M: \( (12, -8) \)
Step 2: Connect the Vertices
Once you have plotted these points on a coordinate plane, connect the vertices in order (J to K to L to M and back to J) to form the quadrilateral JKLM.
Step 3: Find and Plot the Dilated Vertices
Next, we will find the coordinates of the dilated image of each vertex using the dilation formula:
\[ (x', y') = (kx, ky) \]
Where \( k = 0.25 \).
Dilated Coordinates:
- J': \( (0.25 \times 1, 0.25 \times 0) = (0.25, 0) \)
- K': \( (0.25 \times -8, 0.25 \times 4) = (-2, 1) \)
- L': \( (0.25 \times 0, 0.25 \times -4) = (0, -1) \)
- M': \( (0.25 \times 12, 0.25 \times -8) = (3, -2) \)
Step 4: Plot the Dilated Vertices
Now plot the dilated vertices:
- J': \( (0.25, 0) \)
- K': \( (-2, 1) \)
- L': \( (0, -1) \)
- M': \( (3, -2) \)
Step 5: Connect the Dilated Vertices
Connect the new points in order (J' to K' to L' to M' and back to J') to form the image of the quadrilateral after the dilation.
Conclusion
You will have two quadrilaterals on your graph:
- The original quadrilateral JKLM
- The dilated quadrilateral J'K'L'M'
The original quadrilateral will be larger and have its vertices located farther from the origin compared to the dilated quadrilateral, which will be smaller and closer to the origin.
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