Question

question 7 (multiple - choice worth 1 points)
mn has endpoints located at m(-2, 0) and n(2, 0). it was dilated at a scale factor of 2 from center (2, 0). which statement describes the pre - image?
mn is located at m(0, 0) and n(2, 0) and is half the length of mn.
mn is located at m(0, 0) and n(2, 0) and is twice the length of mn.
mn is located at m(0, 0) and n(4, 0) and is half the length of mn.
mn is located at m(0, 0) and n(4, 0) and is twice the length of mn.

Explanation:

Step1: Recall dilation formula

For a point $(x,y)$ dilated by a scale - factor $k$ from a center of dilation $(a,b)$, the formula is $(x',y')=(a + k(x - a),b + k(y - b))$. Here, $y = y'=0$ and $b = 0$, and the center of dilation is $(2,0)$ with $k = 2$. Let the pre - image point be $(x,0)$ and the image point be $(x',0)$. Then $x'=2+(2)(x - 2)$.

Step2: Find pre - image of $M'(-2,0)$

Substitute $x'=-2$ into $x'=2+(2)(x - 2)$.
\[

$$\begin{align*} -2&=2 + 2x-4\\ -2&=2x - 2\\ 2x&=0\\ x&=0 \end{align*}$$

\]

Step3: Find pre - image of $N'(2,0)$

Substitute $x' = 2$ into $x'=2+(2)(x - 2)$.
\[

$$\begin{align*} 2&=2+2x - 4\\ 2&=2x - 2\\ 2x&=4\\ x&=2 \end{align*}$$

\]

Step4: Analyze length relationship

The length of $\overline{M'N'}$ with $M'(-2,0)$ and $N'(2,0)$ is $|2-(-2)| = 4$. The length of $\overline{MN}$ with $M(0,0)$ and $N(2,0)$ is $|2 - 0|=2$. The pre - image $\overline{MN}$ is half the length of the image $\overline{M'N'}$.

Answer:

$\overline{MN}$ is located at $M(0,0)$ and $N(2,0)$ and is half the length of $\overline{M'N'}$.