To determine the effect of the dilation on the angle ABC with a scale factor of 9 and a center point of dilation at the origin (0, 0), we will apply the dilation transformation to the points A, B, and C.
For dilation of a point \((x, y)\) with a scale factor \(k\) centered at the origin, the new coordinates \((x', y')\) are given by: \[ (x', y') = (kx, ky) \]
Given the points:
- \(A(-4, 5)\)
- \(B(-1, 0)\)
- \(C(-3, -3)\)
Applying the dilation with a scale factor of 9:
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For point A: \[ A' = (9 \cdot -4, 9 \cdot 5) = (-36, 45) \]
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For point B: \[ B' = (9 \cdot -1, 9 \cdot 0) = (-9, 0) \]
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For point C: \[ C' = (9 \cdot -3, 9 \cdot -3) = (-27, -27) \]
After performing the dilations, we find the resulting points:
- \(A'(-36, 45)\)
- \(B'(-9, 0)\)
- \(C'(-27, -27)\)
The properties of the dilation apply as follows:
- The angle measurement remains unchanged.
- The corresponding lines (rays) remain parallel between the original angle and the dilated angle.
- The angle is nine times farther from the point of dilation (the origin in this case).
Thus, the correct response is:
After dilation, the angle is nine 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′(−36,45), B′(−9,0), and C′(−27,−27).