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AB is dilated by a scale factor of 4, with a center of dilation at A If A is at point 2,4 and B is at point 5,3, where would A' be located?

Would it be 8,16?

3 answers

To find the coordinates of point A' after dilation of point A with a scale factor of 4 and with the center of dilation at point A, we can follow these steps:

  1. Identify the Coordinates:

    • Point A is at (2, 4).
    • Point B is at (5, 3).

  2. Understanding Dilation with a Center: When a point is dilated from another point, the coordinates of the dilated point can be calculated using the formula:

    \[ A' = A + k(B - A) \]

    where \( k \) is the scale factor, \( A \) is the original point (center of dilation), and \( B \) is the point being dilated.

  3. Substituting the Points: Here, \( A = (2, 4) \) and we are applying a dilation to point A itself, so the target point \( B \) in this case is A still. Therefore, we calculate:

    \[ A' = (2, 4) + 4 \times ((2, 4) - (2, 4)) \]

    Simplifying that yields:

    \[ A' = (2, 4) + 4 \times (0, 0) = (2, 4) + (0, 0) = (2, 4) \]

Thus, point A' is actually located at the same coordinates as A, which is \( (2, 4) \).

The answer you provided, \( (8, 16) \), would be incorrect because it seems to be applying the scale factor to point A directly rather than understanding that point A' coincides with point A during a dilation centered at A.

So, to summarize, the coordinates for A' after dilation are:

A' = (2, 4).

If point A 2,4 had not been the center of dilation would the correct answer have been 8,16?

Yes, if point A (2, 4) was not the center of dilation and you were dilating point B (5, 3) with A as the center of dilation and a scale factor of 4, then you would compute A' as follows:

  1. Identify the Points:

    • Point A = (2, 4)
    • Point B = (5, 3)

  2. Use the Dilation Formula: The formula for finding the dilated point A' from point B using point A as the center of dilation with a scale factor \( k \) is:

    \[ A' = A + k(B - A) \]

  3. Substituting Values: Here, \( k = 4 \), and substituting in the coordinates:

    \[ B - A = (5, 3) - (2, 4) = (5 - 2, 3 - 4) = (3, -1) \]

    Now apply the scale factor:

    \[ A' = (2, 4) + 4 \times (3, -1) \]

    Calculating the scaled vector:

    \[ 4 \times (3, -1) = (12, -4) \]

    Now, add this to A:

    \[ A' = (2, 4) + (12, -4) = (2 + 12, 4 - 4) = (14, 0) \]

So, if A were not the center of dilation, A' would be located at (14, 0), not (8, 16).

Conclusion: \( (8, 16) \) is not the outcome of the dilation in this case. If you were dilating point A itself with A as the center, the coordinates would remain \( (2, 4) \).