Question

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. \\(3sqrt{5}\\) units b. \\(sqrt{3}\\) units c. \\(\frac{sqrt{5}}{3}\\) units d. \\(sqrt{5}\\) units

Explanation:

Step1: Identify coordinates of D, E

From the graph: $D=(0,9)$, $E=(3,3)$

Step2: Calculate length of DE

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$\text{Length of } DE = \sqrt{(3-0)^2+(3-9)^2} = \sqrt{3^2+(-6)^2} = \sqrt{9+36} = \sqrt{45} = 3\sqrt{5}$

Step3: Find length of D'E'

Dilate by scale factor $\frac{1}{3}$: $\text{Length of } D'E' = \frac{1}{3} \times \text{Length of } DE$
$\text{Length of } D'E' = \frac{1}{3} \times 3\sqrt{5} = \sqrt{5}$

Answer:

D. $\sqrt{5}$ units