triangle def is dilated with respect to the origin by a scale factor of $\frac{1}{3}$ to produce $\triangle def$. what is the length of side $de$?
a. $\sqrt{5}$ units
b. $3\sqrt{5}$ units
c. $\frac{\sqrt{5}}{3}$ units
d. $\sqrt{3}$ units

Question

Accepted Answer

# Explanation:
## Step1: Find the coordinates of points D and E
From the graph, $D=(0,9)$ and $E=(2,3)$.
## Step2: Calculate the length of DE using the distance formula
The distance formula is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $D=(0,9)$ and $E=(2,3)$, $x_1 = 0,y_1 = 9,x_2 = 2,y_2 = 3$.
$$
\begin{align*}
DE&=\sqrt{(2 - 0)^2+(3 - 9)^2}\
&=\sqrt{2^2+(- 6)^2}\
&=\sqrt{4 + 36}\
&=\sqrt{40}\
&=2\sqrt{10}
\end{align*}
$$
## Step3: Apply the dilation formula
Since the scale factor $k=\frac{1}{3}$, and $D'E'=k\times DE$.
$$
\begin{align*}
D'E'&=\frac{1}{3}\times DE\
&=\frac{1}{3}\times3\sqrt{5}\
&=\sqrt{5}
\end{align*}
$$
# Answer:
A. $\sqrt{5}$ units

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QUESTION IMAGE

Question

Explanation:

Step1: Find the coordinates of points D and E

Step2: Calculate the length of DE using the distance formula

Step3: Apply the dilation formula

Answer: