2. examine the figure to the right. what angle(s) could the figure be rotated by so the shape does not appear to change?

3. label the vertices (points) on each figure below. then, perform the given transformation. label the vertices of the new transformed figure using prime notation (a’, b’, etc).

a. reflect figure a across line ( l ).

b. rotate figure b ( 90^circ ) clockwise (↻) about point ( p ).

c. reflect figure c across line ( m ).

d. rotate figure d ( 180^circ ) about point ( q ).

3. estimate the measu...

Question

2. examine the figure to the right. what angle(s) could the figure be rotated by so the shape does not appear to change?

3. label the vertices (points) on each figure below. then, perform the given transformation. label the vertices of the new transformed figure using prime notation (a’, b’, etc).

a. reflect figure a across line ( l ).

b. rotate figure b ( 90^circ ) clockwise (↻) about point ( p ).

c. reflect figure c across line ( m ).

d. rotate figure d ( 180^circ ) about point ( q ).

3. estimate the measu...

Accepted Answer

### Question 2 Solution:
# Explanation:
## Step1: Analyze the figure's symmetry
The figure has 4 - fold rotational symmetry (since it looks the same when rotated by 90°, 180°, 270°, and 360°). Let's check the angles:
- A full rotation is $360^\circ$. For a figure with 4 - fold symmetry, the angle of rotation for which it maps onto itself is $\frac{360^\circ}{n}$, where $n$ is the number of times it repeats. Here, $n = 4$, so $\frac{360^\circ}{4}=90^\circ$. Also, multiples of $90^\circ$ (like $180^\circ$, $270^\circ$, $360^\circ$) will also work because rotating by $180^\circ$ (which is $2\times90^\circ$) or $270^\circ$ (which is $3\times90^\circ$) or $360^\circ$ (which is $4\times90^\circ$) will still make the figure look the same.

## Step2: Confirm the angles
The figure is symmetric about 90° rotations. So the angles are $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$. But the most basic non - trivial angle is $90^\circ$ (and its multiples). Since the question asks "what angle(s)", the key angles are $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$. But typically, for such a symmetric figure (like a 4 - fold symmetric figure), the smallest non - zero angle of rotation that maps it onto itself is $90^\circ$, and also $180^\circ$, $270^\circ$, $360^\circ$.

# Answer:
The figure can be rotated by $90^\circ$, $180^\circ$, $270^\circ$, or $360^\circ$ (the most common answer focuses on $90^\circ$, $180^\circ$, and $270^\circ$ as non - trivial, with $360^\circ$ being a full rotation). If we consider the smallest non - zero angle, $90^\circ$ (and its multiples $180^\circ$, $270^\circ$, $360^\circ$).

### Question 3a Solution:
# Explanation:
## Step1: Label vertices of Figure A
Let's assume the vertices of Figure A (the triangle) are labeled as $A_1$, $A_2$, $A_3$ (you can label them based on their positions, e.g., top vertex, bottom vertex, left vertex).
## Step2: Reflect across line $l$
To reflect a point $(x,y)$ across a vertical line $x = a$ (assuming line $l$ is vertical, from the grid), the rule is $(x,y)\to(2a - x,y)$. For each vertex of Figure A, apply this reflection rule. For example, if a vertex is at $(x,y)$ and line $l$ is at $x = k$, the reflected vertex $A_1'$ will be at $(2k - x,y)$.
## Step3: Label the new vertices
After reflecting each vertex, label them as $A'$, $B'$, $C'$ (using prime notation) corresponding to the original vertices $A$, $B$, $C$ of Figure A.

# Answer:
1. Label the original vertices of Figure A (e.g., let the triangle have vertices $A$, $B$, $C$ with coordinates based on the grid).
2. For each vertex, reflect across line $l$ using the reflection formula (for vertical line $x = a$, $(x,y)\to(2a - x,y)$; for non - vertical, use the appropriate reflection formula for a line).
3. Label the reflected vertices as $A'$, $B'$, $C'$ (or other prime - labeled vertices based on original labels).

### Question 3b Solution:
# Explanation:
## Step1: Label vertices of Figure B
Label the vertices of Figure B (the polygon) as $B_1$, $B_2$, $B_3$, $B_4$ (or other labels) with respect to point $P$.
## Step2: Rotate $90^\circ$ clockwise about $P$
The rotation rule for a point $(x,y)$ about a center $(h,k)$ by $90^\circ$ clockwise is $(x - h,y - k)\to(y - k,-(x - h))$, then add back $(h,k)$ to get $(h+(y - k),k-(x - h))=(h + y - k,-x + h + k)$. For each vertex of Figure B, apply this rotation rule with center $P$'s coordinates $(h,k)$.
## Step3: Label the new vertices
After rotating each vertex, label them as $B'$, $C'$, $D'$, $E'$ (using prime notation) corresponding to the original vertices of Figure B.

# Answer:
1. Label the original vertices of Figure B (e.g., vertices $B$, $C$, $D$, $E$) with respect to point $P$.
2. For each vertex, apply the $90^\circ$ clockwise rotation formula about point $P$: if a vertex is $(x,y)$ and $P=(h,k)$, the new vertex is $(h+(y - k),k-(x - h))$.
3. Label the rotated vertices as $B'$, $C'$, $D'$, $E'$ (or other prime - labeled vertices based on original labels).

### Question 3c Solution:
# Explanation:
## Step1: Label vertices of Figure C
Label the vertices of Figure C (the polygon) as $C_1$, $C_2$, $C_3$, $C_4$, $C_5$ (or other labels) with respect to line $m$.
## Step2: Reflect across line $m$
The reflection across a line $y = mx + b$ can be done using the reflection formula. For a line with slope $m = 1$ (like a diagonal line $y=x + c$), the reflection of a point $(x,y)$ is $(y - c,x + c)$ (adjusting for the intercept). For each vertex of Figure C, apply the reflection formula for line $m$.
## Step3: Label the new vertices
After reflecting each vertex, label them as $C'$, $D'$, $E'$, $F'$, $G'$ (using prime notation) corresponding to the original vertices of Figure C.

# Answer:
1. Label the original vertices of Figure C (e.g., vertices $C$, $D$, $E$, $F$, $G$) with respect to line $m$.
2. For each vertex, apply the reflection formula across line $m$ (depending on the line's slope and intercept).
3. Label the reflected vertices as $C'$, $D'$, $E'$, $F'$, $G'$ (or other prime - labeled vertices based on original labels).

### Question 3d Solution:
# Explanation:
## Step1: Label vertices of Figure D
Label the vertices of Figure D (the polygon) as $D_1$, $D_2$, $D_3$, $D_4$ (or other labels) with respect to point $Q$.
## Step2: Rotate $180^\circ$ about $Q$
The rotation rule for a point $(x,y)$ about a center $(h,k)$ by $180^\circ$ is $(x - h,y - k)\to(-(x - h),-(y - k))$, then add back $(h,k)$ to get $(h-(x - h),k-(y - k))=(2h - x,2k - y)$. For each vertex of Figure D, apply this rotation rule with center $Q$'s coordinates $(h,k)$.
## Step3: Label the new vertices
After rotating each vertex, label them as $D'$, $E'$, $F'$, $G'$ (using prime notation) corresponding to the original vertices of Figure D.

# Answer:
1. Label the original vertices of Figure D (e.g., vertices $D$, $E$, $F$, $G$) with respect to point $Q$.
2. For each vertex, apply the $180^\circ$ rotation formula about point $Q$: if a vertex is $(x,y)$ and $Q=(h,k)$, the new vertex is $(2h - x,2k - y)$.
3. Label the rotated vertices as $D'$, $E'$, $F'$, $G'$ (or other prime - labeled vertices based on original labels).

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QUESTION IMAGE

Question

Explanation:

Question 2 Solution:

Step1: Analyze the figure's symmetry

Step2: Confirm the angles

Step1: Label vertices of Figure A

Step2: Reflect across line \(l\)

Step3: Label the new vertices

Step1: Label vertices of Figure B

Step2: Rotate \(90^\circ\) clockwise about \(P\)

Step3: Label the new vertices

Answer:

Question 3a Solution: