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 1 comma 2 right parenthesis and upper B prime is at left parenthesis 3 comma 3 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 (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,−2) and B′ is at (−3,−3) .

Answer 1

To prove that the length of the segment \( A'B' \) is half the length of \( AB \) after a dilation about the origin with a scale factor of 12, we need to first clarify what a dilation does to a point in the plane.

For a point \( (x, y) \) under a dilation centered at the origin with a scale factor \( k \), the new point \( (x', y') \) is given by: \[ (x', y') = (kx, ky) \]

In your case, the scale factor is 12. Therefore: \[ (x', y') = (12x, 12y) \]

Now, if we denote the coordinates of points \( A \) and \( B \) as \( A(x_1, y_1) \) and \( B(x_2, y_2) \), their dilated counterparts \( A' \) and \( B' \) will be: \[ A' = (12x_1, 12y_1) \] \[ B' = (12x_2, 12y_2) \]

Next, we need to check which of the provided options for \( A' \) and \( B' \) yield a segment \( A'B' \) that is half the length of \( AB \).

Let's compute the length of \( AB \): \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

And the length of \( A'B' \): \[ A'B' = \sqrt{(12x_2 - 12x_1)^2 + (12y_2 - 12y_1)^2} = \sqrt{144((x_2 - x_1)^2 + (y_2 - y_1)^2)} = 12 \cdot AB \]

This shows us that under a dilation with a scale factor of 12, the segment \( A'B' \) is 12 times longer than \( AB \), not half.

To find \( A' \) and \( B' \) corresponding to having \( AB = 2A'B' \), the scale factor would need to be \( \frac{1}{2} \), not 12.

Thus, the correct points \( A' \) and \( B' \) must be such that they are at a scale factor of \( \frac{1}{2} \) when dilating from the points \( A \) and \( B \). If we were scaling down by half, we would need their coordinates to be: \[ A' = (0.5x_1, 0.5y_1) \] \[ B' = (0.5x_2, 0.5y_2) \]

Unfortunately, instead of this, it appears the options provided do not reflect a situation where \( A'B' \) is indeed half of \( AB \).

None of the responses provided correctly relate the segments \( A'B' \) and \( AB \) according to the stipulated geometric transformations.

If checking against the options given, none match with the dilation required to create \( A' \) and \( B' \) as half-length compared to \( AB \).

To summarize, since the intended goal was to show that \( A'B' \) is half the distance of \( AB \), and by performing the dilation where \( k = 12 \) produced a length \( 12 \times AB \), it contradicts the claim that \( A'B' \) is half of \( AB \). Therefore, there may be a misunderstanding in the dilation statement as originally posed.