To analyze the effect of a dilation on angle ABC with points A(-5, 3), B(-4, -2), and C(1, 4), with a scale factor of 8 centered at the origin (0, 0), we can apply the properties of dilation.
Steps for Dilation
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Dilation Formula: The new coordinates of a point (x, y) after dilation centered at the origin with a scale factor \(k\) will be: \[ (kx, ky) \] In this case, \(k = 8\).
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Calculate New Coordinates:
- For point A(-5, 3): \[ A' = (8 \cdot -5, 8 \cdot 3) = (-40, 24) \]
- For point B(-4, -2): \[ B' = (8 \cdot -4, 8 \cdot -2) = (-32, -16) \]
- For point C(1, 4): \[ C' = (8 \cdot 1, 8 \cdot 4) = (8, 32) \]
Properties of Dilation
- The angle measurement remains the same after dilation.
- The rays forming the angle are parallel between the original angle and the dilated angle.
- The points A', B', and C' are indeed eight times farther from the origin than points A, B, and C.
Conclusion
The correct description of the dilation of angle ABC with the given points A', B', and C' is:
After dilation, the angle is eight times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are A′(−40,24), B′(−32,−16), and C′(8,32).
So the response would be:
- "After dilation, the angle is eight times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are A′(−40,24), B′(−32,−16), and C′(8,32)."