Question

determine whether trapezoids defg and lmnp with the given vertices are similar. use transformations to explain your reasoning.
d(-4, 3), e(-2, 3), f(-1, 1), g(-4, 1) and l(1, -1), m(3, -1), n(6, -3), p(1, -3)

• yes; defg can be mapped to lmnp by a dilation with a scale factor of \\(\frac{5}{3}\\) followed by a translation 4 units right and 4 units down.
• yes; defg can be mapped to lmnp by a reflection over the x - axis followed by a dilation with a scale factor of \\(\frac{5}{3}\\), followed by a translation 4 units down.
• no; the scale factor from \\(\overline{dg}\\) to \\(\overline{lp}\\) is 2, but the scale factor from \\(\overline{fg}\\) to \\(\overline{np}\\) is \\(\frac{2}{3}\\).
• no; the scale factor from \\(\overline{dg}\\) to \\(\overline{lp}\\) is 1, but the scale factor from \\(\overline{fg}\\) to \\(\overline{np}\\) is \\(\frac{5}{3}\\).

Explanation:

Step1: Calculate length of $\overline{DG}$

Use distance formula: $DG = |3-1| = 2$

Step2: Calculate length of $\overline{LP}$

Use distance formula: $LP = |-1-(-3)| = 2$
Wait, correction: $DG$ is vertical: $D(-4,3), G(-4,1)$ so $DG = 3-1=2$. $L(1,-1), P(1,-3)$ so $LP = -1-(-3)=2$. Scale factor $\frac{LP}{DG}=\frac{2}{2}=1$.

Step3: Calculate length of $\overline{FG}$

$F(-1,1), G(-4,1)$: $FG = |-1 - (-4)| = 3$

Step4: Calculate length of $\overline{NP}$

$N(6,-3), P(1,-3)$: $NP = |6-1| = 5$

Step5: Find scale factors

Scale factor $\frac{NP}{FG}=\frac{5}{3}$. Scale factors for corresponding sides are not equal ($1
eq \frac{5}{3}$), so trapezoids are not similar.
Wait, correction to match option: Recheck $\overline{DG}$ and $\overline{LP}$ as vertical legs, $\overline{FG}$ and $\overline{NP}$ as top/bottom bases.
Wait, $DEFG$: $DE$ is top base: $D(-4,3), E(-2,3)$ length $2$; $FG$ is bottom base: $3$. $LMNP$: $LM$ is top base: $L(1,-1), M(3,-1)$ length $2$; $NP$ is bottom base: $5$. Scale factor for top bases $\frac{2}{2}=1$, scale factor for bottom bases $\frac{5}{3}$. The correct option is the one stating unequal scale factors, which is: no; The scale factor from $\overline{DG}$ to $\overline{LP}$ is 1, but the scale factor from $\overline{FG}$ to $\overline{NP}$ is $\frac{5}{3}$. Wait, recheck:
$DG$ length: $3-1=2$, $LP$ length: $-1 - (-3)=2$, so scale factor $\frac{LP}{DG}=1$.
$FG$ length: $|-1 - (-4)|=3$, $NP$ length: $|6-1|=5$, so scale factor $\frac{NP}{FG}=\frac{5}{3}$.
The correct option is: no; The scale factor from $\overline{DG}$ to $\overline{LP}$ is 1, but the scale factor from $\overline{FG}$ to $\overline{NP}$ is $\frac{5}{3}$.

Corrected Explanation:

Step1: Compute $\overline{DG}$ length

$DG = |3-1| = 2$

Step2: Compute $\overline{LP}$ length

$LP = |-1-(-3)| = 2$

Step3: Find $\frac{LP}{DG}$ scale factor

$\frac{LP}{DG} = \frac{2}{2} = 1$

Step4: Compute $\overline{FG}$ length

$FG = |-1-(-4)| = 3$

Step5: Compute $\overline{NP}$ length

$NP = |6-1| = 5$

Step6: Find $\frac{NP}{FG}$ scale factor

$\frac{NP}{FG} = \frac{5}{3}$

Step7: Compare scale factors

$1
eq \frac{5}{3}$, so trapezoids not similar.

Final Answer: no; The scale factor from $\overline{DG}$ to $\overline{LP}$ is 1, but the scale factor from $\overline{FG}$ to $\overline{NP}$ is $\frac{5}{3}$.

Answer:

no; The scale factor from $\overline{DG}$ to $\overline{LP}$ is 2, but the scale factor from $\overline{FG}$ to $\overline{NP}$ is $\frac{2}{3}$.