Question

are the two figures similar? if so, describe a sequence of transformations that maps one figure onto the other figure. if not, explain why they are not similar.
choose the correct answer and justification.
similar; dilate triangle key using a scale factor of 2 and center of dilation at the origin, then rotate it 90° clockwise about the origin, and translate it 1 unit left and 3 units down.
similar; dilate triangle key using a scale factor of 2 and center of dilation at (-2, 2), then rotate it 90° clockwise about (-1, 1), and translate it 1 unit right and 2 units up.
not similar; in triangle key, side \\(\overline{ey}\\) has a length of 2.8 units. the corresponding side in triangle fob has a length of 5.6 units.
not similar; the ratios of the side lengths are not the same for sides \\(\overline{ky}\\) and \\(\overline{ey}\\).

Explanation:

Step1: Check side length ratios

First, identify the lengths of corresponding sides:

• $\overline{EY} = 2.8$ units, corresponding $\overline{FB} = 5.6$ units. The ratio is $\frac{5.6}{2.8}=2$.
• $\overline{KY}$ is a vertical side of 2 units, its corresponding vertical side $\overline{FO}$ is 4 units. Ratio: $\frac{4}{2}=2$.
• $\overline{KE}$ is a horizontal side of 2 units, its corresponding horizontal side $\overline{OB}$ is 4 units. Ratio: $\frac{4}{2}=2$.

All side ratios are equal, so figures are similar.

Step2: Verify transformation sequence

Check the first option: Dilate $\triangle KEY$ by scale factor 2 at origin:

• $K(-3,1)\to K'(-6,2)$, $E(-1,1)\to E'(-2,2)$, $Y(-3,3)\to Y'(-6,6)$

Rotate $90^\circ$ clockwise about origin: $(x,y)\to(y,-x)$

• $K'(-6,2)\to K''(2,6)$, $E'(-2,2)\to E''(2,2)$, $Y'(-6,6)\to Y''(6,6)$

Translate 1 unit left, 3 units down: $(x,y)\to(x-1,y-3)$

• $K''(2,6)\to(1,3)$, $E''(2,2)\to(1,-1)$, $Y''(6,6)\to(5,3)$ which matches $F(1,3)$, $O(1,-1)$, $B(5,3)$.

Step3: Eliminate incorrect options

• Second option: The transformation results do not match $\triangle FOB$.
• Third option: Incorrectly claims figures are not similar (ratios are equal).
• Fourth option: Incorrect, ratios of $\overline{KY}$ and $\overline{EY}$ are $\frac{2}{2.8}=\frac{5}{7}$, and their corresponding sides have the same ratio $\frac{4}{5.6}=\frac{5}{7}$.

Answer:

similar; Dilate triangle KEY using a scale factor of 2 and center of dilation at the origin, then rotate it 90° clockwise about the origin, and translate it 1 unit left and 3 units down.