ab has endpoints at (-1, -1) and (-2, -4). its transformation ab has endpoints at (-2, -2) and (-4, -8). what type of transformation occurred? are lengths of segments affected by this type of transformation? explain.
a rotation; no. the lengths of the two segments are the same.
b reflection; yes. the length of segment ab is longer than the original segment ab.
c dilation by a scale factor of 2; yes. the length of segment ab is longer than the original segment ab.
d dilation by a scale factor of $ \frac { 1 } { 2 } $. the length of segment ab is shorter than the original segment ab.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the length of segment $AB$
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $A(-1,-1)$ and $B(-2,-4)$, $x_1=-1,y_1 = - 1,x_2=-2,y_2=-4$.
$$
\begin{align*}
AB&=\sqrt{(-2 + 1)^2+(-4 + 1)^2}\
&=\sqrt{(-1)^2+(-3)^2}\
&=\sqrt{1 + 9}\
&=\sqrt{10}
\end{align*}
$$
## Step2: Calculate the length of segment $A'B'$
For $A'(-2,-2)$ and $B'(-4,-8)$, $x_1=-2,y_1=-2,x_2=-4,y_2=-8$.
$$
\begin{align*}
A'B'&=\sqrt{(-4 + 2)^2+(-8 + 2)^2}\
&=\sqrt{(-2)^2+(-6)^2}\
&=\sqrt{4+36}\
&=\sqrt{40}=2\sqrt{10}
\end{align*}
$$
## Step3: Analyze the transformation
We can see that $(x,y)\to(2x,2y)$. This is a dilation by a scale factor of $2$. Since $A'B' = 2AB$, the length of the segment is affected.

# Answer:
C. dilation by a scale factor of $2$; yes. The length of segment $A'B'$ is longer than the original segment $AB$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of segment \(AB\)

Step2: Calculate the length of segment \(A'B'\)

Step3: Analyze the transformation

Answer: