Dilations of Segments Quick Check 3 of 53 of 5 Items Question Use the image to answer the question. A dilation of AB¯¯¯¯¯¯¯¯ occurs using a scale factor of 12 and a center of dilation at the origin. Prove that A′B′¯¯¯¯¯¯¯¯¯¯ is half the length of AB¯¯¯¯¯¯¯¯ by finding the points A′ and B′ . (1 point) Responses A′ is at (−1,−2) and B′ is at (−3,−3). upper A prime is at left parenthesis negative 1 comma negative 2 right parenthesis and upper B prime is at left parenthesis negative 3 comma negative 3 right parenthesis . A′ is at (4,8) and B′ is at (12,12). upper A prime is at left parenthesis 4 comma 8 right parenthesis and upper B prime is at left parenthesis 12 comma 12 right parenthesis . A′ is at (1,5) and B′ is at (3,6). upper A prime is at left parenthesis 1 comma 5 right parenthesis and upper B prime is at left parenthesis 3 comma 6 right parenthesis . A′ is at (1,2) and B′ is at (3,3). upper A prime is at left parenthesis 1 comma 2 right parenthesis and upper B prime is at left parenthesis 3 comma 3 right parenthesis . Skip to navigation page 15 of 15 Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.

Answer 1

To find points \( A' \) and \( B' \) that are a result of a dilation of segment \( \overline{AB} \) using a scale factor of \( \frac{1}{2} \) (which is the same as halving the length of \( \overline{AB} \)), with the center of dilation at the origin, follow these steps:

1. Identify Points A and B: Although the question does not specify points \( A \) and \( B \), let's assume they are associated with the segments provided in the responses.

2. Dilation Formula: The dilation of a point \( (x, y) \) with respect to the origin and a scale factor \( k \) is given by: \[ (x', y') = (k \cdot x, k \cdot y) \] In this case, since we need to prove that \( A'B' \) is half the length of \( AB \), we will use \( k = \frac{1}{2} \).

3. Apply Dilation to Point A: If we take generic coordinates for points \( A \) and \( B \), we can illustrate:

   • Let \( A = (x_A, y_A) \)
   • Then, \( A' = \left(\frac{1}{2}x_A, \frac{1}{2}y_A\right) \)

4. Apply Dilation to Point B: Similarly:

   • Let \( B = (x_B, y_B) \)
   • Then, \( B' = \left(\frac{1}{2}x_B, \frac{1}{2}y_B\right) \)

5. Check the Options:

   • A dilation with a scale factor of 12 would make the lengths larger, meaning no option can provide results that show \( A'B' \) is actually half the length of \( AB \).

   Given the choices:

   • \( A' = (-1, -2) \) and \( B' = (-3, -3) \)
   • \( A' = (4, 8) \) and \( B' = (12, 12) \)
   • \( A' = (1, 5) \) and \( B' = (3, 6) \)
   • \( A' = (1, 2) \) and \( B' = (3, 3) \)

   The only option that would reflect a dilation that doesn't increase would be fractional coordinates (but these seem mapped back incorrectly).

6. Conclusion: You need to check where your points \( A \) and \( B \) are coming from in the diagram, analyze them carefully, and calculate using the appropriate scale factor.

Thus, without direct coordinates for \( A \) and \( B \), we cannot determine \( A' \) and \( B' \) completely accurately from the options provided. If additional clarification for \( A \) and \( B \) is retrieved, apply the described method to arrive at the correct dilated points.