Question

a. △cma is a rotation of △lah about the origin. which statement must be true? a. the angle measures are preserved, but the side - lengths are reduced. b. the angle measures and side lengths are preserved. c. the angle of rotation is 90° clockwise. d. the angle of rotation is 90° counterclockwise.
b. a. what scale factor must i use to change $overrightarrow{ec}$ to $overrightarrow{ah}$?__ b. what scale factor must i use to change $overrightarrow{eh}$ to $overrightarrow{fg}$? c. what scale factor must i use to change $overrightarrow{hf}$ to $overrightarrow{hb}$?__
c. complete the statement about the dilation. from each pair, select the word that is correct. if the figures are similar, the corresponding sides are (congruent, proportional), and the corresponding angles are (congruent, proportional).

Explanation:

Step 1: Recall properties of rotation

A rotation is a rigid - motion transformation. Rigid - motion transformations preserve both angle measures and side lengths.

Step 2: Analyze each option for part a

• Option a: Side lengths are not reduced in a rotation, so this is false.
• Option b: Since rotation is a rigid - motion, angle measures and side lengths are preserved. This is true.
• Option c: There is no information given to suggest the angle of rotation is 90° clockwise, so this is not necessarily true.
• Option d: There is no information given to suggest the angle of rotation is 90° counter - clockwise, so this is not necessarily true.

Step 3: For part b (a)

Count the number of units between points on $\overrightarrow{EC}$ and $\overrightarrow{AH}$. Let's assume each grid unit is 1. If we count the length of $\overrightarrow{EC}$ and $\overrightarrow{AH}$, and find the ratio. Suppose $EC = x$ and $AH=y$, the scale factor $k=\frac{y}{x}$. Without specific grid - based length measurements (assuming from the grid), if $EC = 3$ and $AH = 9$, then the scale factor $k = 3$.

Step 4: For part b (b)

Similarly, count the lengths of $\overrightarrow{EH}$ and $\overrightarrow{FG}$. Let the length of $\overrightarrow{EH}=m$ and $\overrightarrow{FG}=n$. The scale factor is $\frac{n}{m}$. If $EH = 6$ and $FG = 2$, then the scale factor is $\frac{1}{3}$.

Step 5: For part b (c)

Count the lengths of $\overrightarrow{HF}$ and $\overrightarrow{HB}$. Let the length of $\overrightarrow{HF}=p$ and $\overrightarrow{HB}=q$. The scale factor is $\frac{q}{p}$. If $HF = 4$ and $HB = 8$, then the scale factor is 2.

Step 6: For part c

If the figures are similar, the corresponding angles are congruent and the corresponding sides are proportional.

Answer:

a. b. The angle measures and side lengths are preserved.
b. (a) [scale factor value based on grid - counting]
(b) [scale factor value based on grid - counting]
(c) [scale factor value based on grid - counting]
c. congruent; proportional