Bot GPT 4-o mini To find the magnitude of dilation from the original parallelogram ABCD to the dilated parallelogram A'B'C'D', we can use the formula for dilation. If a point \( P(x, y) \) is dilated to a point \( P'(x', y') \) in reference to the origin with a scale factor \( k \), then:
\[ P' = k \cdot P \]
For each point, we can calculate the scale factor \( k \) by taking the ratio of the coordinates of the dilated point to the original point.
Let's calculate the scale factor for each point.
-
For point A (-2, 4) to A' (-1, 2): \[ k_A = \frac{x'}{x} = \frac{-1}{-2} = \frac{1}{2} \] \[ k_A = \frac{y'}{y} = \frac{2}{4} = \frac{1}{2} \] Both x and y give the same scale factor \( k_A = \frac{1}{2} \).
-
For point B (4, 4) to B' (2, 2): \[ k_B = \frac{x'}{x} = \frac{2}{4} = \frac{1}{2} \] \[ k_B = \frac{y'}{y} = \frac{2}{4} = \frac{1}{2} \] Both x and y give the same scale factor \( k_B = \frac{1}{2} \).
-
For point C (2, -2) to C' (1, -1): \[ k_C = \frac{x'}{x} = \frac{1}{2} = \frac{1}{2} \] \[ k_C = \frac{y'}{y} = \frac{-1}{-2} = \frac{1}{2} \] Both x and y give the same scale factor \( k_C = \frac{1}{2} \).
-
For point D (-4, -2) to D' (-2, -1): \[ k_D = \frac{x'}{x} = \frac{-2}{-4} = \frac{1}{2} \] \[ k_D = \frac{y'}{y} = \frac{-1}{-2} = \frac{1}{2} \] Both x and y give the same scale factor \( k_D = \frac{1}{2} \).
Conclusion
In all cases, the scale factor \( k \) is consistent:
\[ k = \frac{1}{2} \]
Thus, the magnitude of dilation for parallelogram ABCD to parallelogram A'B'C'D' is \( \frac{1}{2} \).
3 answers