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 (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)
.
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,−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 (1,5)
and B′
is at (3,6)
.

Answer 1

To prove that \( A'B' \) is half the length of \( AB \) after a dilation of \( AB \) with a scale factor of 12 and a center of dilation at the origin, we first need to find the new points \( A' \) and \( B' \) based on their respective original coordinates \( A(x_A, y_A) \) and \( B(x_B, y_B) \).

The formula for dilation is given by:
\[
A' = k \cdot A
\]
\[
B' = k \cdot B
\]
where \( k \) is the scale factor and \( A' \) and \( B' \) are the new coordinates after dilation.

Given:
- Scale factor \( k = \frac{1}{12} \)
- Center of dilation at the origin (0, 0)

Let's assume original points \( A \) and \( B \) are \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \).

After dilation by scale factor of \(\frac{1}{12}\):

\[
A' = \left( \frac{x_A}{12}, \frac{y_A}{12} \right)
\]
\[
B' = \left( \frac{x_B}{12}, \frac{y_B}{12} \right)
\]

Next, let's calculate the length \( AB \) and \( A'B' \):

1. Distance between points \( A \) and \( B \):
\[
AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\]

2. Distance between points \( A' \) and \( B' \):
\[
A'B' = \sqrt{\left( \frac{x_B}{12} - \frac{x_A}{12} \right)^2 + \left( \frac{y_B}{12} - \frac{y_A}{12} \right)^2}
\]
\[
A'B' = \sqrt{\frac{(x_B - x_A)^2}{144} + \frac{(y_B - y_A)^2}{144}} = \frac{1}{12}\sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \frac{1}{12}AB
\]

Therefore, since we have a scale factor of \( \frac{1}{12} \), the conclusion is:
\[
A'B' = \frac{1}{12}AB
\]
So to answer the question, none of the choices presented seem to give a valid definition that corresponds to this dilation context.

If we want \( A'B' \) to be half the original \( AB \), then we should use a scale factor of \( 2 \).

Therefore, we can conclude that with the given scale factor of \( \frac{1}{12} \), \( A'B' \) will not be half the length of \( AB \).