Question

determine whether $\triangle ghi$ and $\triangle jkl$ with the given vertices are similar. use transformations to explain your reasoning.
$g(-2, 3)$, $h(4, 3)$, $i(4, 0)$ and $j(1, 0)$, $k(6, -2)$, $l(1, -2)$
\bigcirc yes; $\triangle ghi$ can be mapped to $\triangle jkl$ by a dilation with center $(-14, 0)$ and a scale factor of $\frac{2}{3}$ followed by a $180^\circ$ rotation about the origin.
\bigcirc yes; $\triangle ghi$ can be mapped to $\triangle jkl$ by a $180^\circ$ rotation about the origin, a dilation with a scale factor of $\frac{2}{3}$, and a translation 3 units right.
\bigcirc no; the scale factor from $\overline{hi}$ to $\overline{jl}$ is $\frac{2}{3}$, but the scale factor from $\overline{gh}$ to $\overline{kl}$ is $\frac{5}{6}$.
\bigcirc no; the scale factor from $\overline{hi}$ to $\overline{jl}$ is $\frac{3}{2}$, but the scale factor from $\overline{gh}$ to $\overline{kl}$ is $\frac{6}{5}$.

Explanation:

Step1: Calculate side lengths of △GHI

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:

• $GH=\sqrt{(4-(-2))^2+(3-3)^2}=\sqrt{6^2}=6$
• $HI=\sqrt{(4-4)^2+(0-3)^2}=\sqrt{(-3)^2}=3$
• $GI=\sqrt{(4-(-2))^2+(0-3)^2}=\sqrt{6^2+(-3)^2}=\sqrt{45}=3\sqrt{5}$

Step2: Calculate side lengths of △JKL

Use distance formula:

• $JK=\sqrt{(6-1)^2+(-2-0)^2}=\sqrt{5^2+(-2)^2}=\sqrt{29}$
• $KL=\sqrt{(1-6)^2+(-2-(-2))^2}=\sqrt{(-5)^2}=5$
• $JL=\sqrt{(1-1)^2+(-2-0)^2}=\sqrt{(-2)^2}=2$

Step3: Check scale factors

Compare corresponding sides:

• $\frac{JL}{HI}=\frac{2}{3}$
• $\frac{KL}{GH}=\frac{5}{6}$

Since $\frac{2}{3}
eq\frac{5}{6}$, the scale factors are not consistent.

Answer:

no; The scale factor from $\overline{HI}$ to $\overline{JL}$ is $\frac{2}{3}$, but the scale factor from $\overline{GH}$ to $\overline{KL}$ is $\frac{5}{6}$