Question

part e
dilate △abc centered at the origin using the scale factor n from part d.
measure and compare the lengths of the sides of the dilated triangle, △abc, with those of △def. take a screenshot of your dilation, save it, and insert the image below the tables. for help with dilations, watch this short video on how to complete transformations in geogebra.
side length
ab
bc
ac
side length
de
ef
fd
space used(includes formatting): 1653 / 30000

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is dilated centered at the origin with scale - factor $n$, the new point $(x',y')$ is given by $(x',y')=(nx,ny)$. For a triangle $\triangle ABC$ with vertices $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, the vertices of the dilated triangle $\triangle A'B'C'$ are $A'(nx_1,ny_1)$, $B'(nx_2,ny_2)$, $C'(nx_3,ny_3)$.

Step2: Use distance formula to find side - lengths

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Calculate the lengths of $A'B'$, $B'C'$, $A'C'$ using the coordinates of $A'$, $B'$, $C'$ and the lengths of $DE$, $EF$, $FD$ using the coordinates of $D$, $E$, $F$.

Step3: Compare side - lengths

If the scale - factor $n$ is chosen correctly such that $\triangle A'B'C'$ and $\triangle DEF$ are similar (or congruent in a special case), we can analyze the ratios of the corresponding side - lengths. For example, if $\frac{A'B'}{DE}=\frac{B'C'}{EF}=\frac{A'C'}{FD}=k$, we can determine the relationship between the two triangles.

However, since no coordinates of the vertices of $\triangle ABC$ and $\triangle DEF$ are given, we cannot perform the actual calculations. In a practical situation in GeoGebra:

1. Enter the coordinates of the vertices of $\triangle ABC$.
2. Use the dilation tool with the given scale - factor $n$ centered at the origin to get $\triangle A'B'C'$.
3. Use the distance measurement tool to measure the lengths of $A'B'$, $B'C'$, $A'C'$, $DE$, $EF$, $FD$.
4. Compare the measured values.

Answer:

Without the coordinates of the vertices of $\triangle ABC$ and $\triangle DEF$, we cannot provide numerical values for the side - lengths and their comparisons. In practice, use GeoGebra as described above to complete the task.