1. if ( overline{er} ) is rotated about point ( m ) by ( 30^{circ} ), then which of the following is true about the length of ( overline{er} ) compared to the length of its image, ( overline{er} )?
(1) ( er = 2er )
(2) ( er=\frac{1}{2}er )
(3) ( er = er )
(4) ( er = er + 30 )
2. in the image shown, ( overline{hi} ) is the image of ( overline{hi} ) after a clockwise rotation of ( 104^{circ} ) about point ( c ). which of the following distances do not have to be the same?
(1) the distance from ( c ) to ( i ) and the distance from ( c ) to ( h )
(2) the distance from ( c ) to ( h ) and the distance from ( c ) to ( h )
(3) the distance from ( h ) to ( i ) and the distance from ( h ) to ( i )
(4) the distance from ( i ) to ( c ) and the distance from ( c ) to ( i )
3. in the figure shown below, ( riangle qrs ) is the image of ( riangle mnp ) after a clockwise rotation about point ( c ) by an angle of ( 120^{circ} ). the angles of ( riangle mnp ) are shown. which of the following must be the measure of ( angle r )?
(1) ( 51^{circ} )
(2) ( 59^{circ} )
(3) ( 70^{circ} )
(4) ( 120^{circ} )
4. which of the following would rotate a point in the coordinate grid by ( 90^{circ} ) clockwise about the origin?
(1) ( (x,y) o(y,x) )
(3) ( (x,y) o(-x,-y) )
(2) ( (x,y) o(-y,x) )
(4) ( (x,y) o(y,-x) )
5. if ( overline{wx} ) is rotated by ( 90^{circ} ) counterclockwise about the origin, then its image would have endpoints at
(1) ( w(1,2) ) and ( x(4,8) )
(2) ( w(-1,2) ) and ( x(-4,8) )
(3) ( w(2,-1) ) and ( x(8,-4) )
(4) ( w(-2,1) ) and ( x(8,4) )

Question

1. if ( overline{er} ) is rotated about point ( m ) by ( 30^{circ} ), then which of the following is true about the length of ( overline{er} ) compared to the length of its image, ( overline{er} )?
(1) ( er = 2er )
(2) ( er=\frac{1}{2}er )
(3) ( er = er )
(4) ( er = er + 30 )
2. in the image shown, ( overline{hi} ) is the image of ( overline{hi} ) after a clockwise rotation of ( 104^{circ} ) about point ( c ). which of the following distances do not have to be the same?
(1) the distance from ( c ) to ( i ) and the distance from ( c ) to ( h )
(2) the distance from ( c ) to ( h ) and the distance from ( c ) to ( h )
(3) the distance from ( h ) to ( i ) and the distance from ( h ) to ( i )
(4) the distance from ( i ) to ( c ) and the distance from ( c ) to ( i )
3. in the figure shown below, ( 	riangle qrs ) is the image of ( 	riangle mnp ) after a clockwise rotation about point ( c ) by an angle of ( 120^{circ} ). the angles of ( 	riangle mnp ) are shown. which of the following must be the measure of ( angle r )?
(1) ( 51^{circ} )
(2) ( 59^{circ} )
(3) ( 70^{circ} )
(4) ( 120^{circ} )
4. which of the following would rotate a point in the coordinate grid by ( 90^{circ} ) clockwise about the origin?
(1) ( (x,y)	o(y,x) )
(3) ( (x,y)	o(-x,-y) )
(2) ( (x,y)	o(-y,x) )
(4) ( (x,y)	o(y,-x) )
5. if ( overline{wx} ) is rotated by ( 90^{circ} ) counterclockwise about the origin, then its image would have endpoints at
(1) ( w(1,2) ) and ( x(4,8) )
(2) ( w(-1,2) ) and ( x(-4,8) )
(3) ( w(2,-1) ) and ( x(8,-4) )
(4) ( w(-2,1) ) and ( x(8,4) )

Accepted Answer

### Question 1
# Explanation:
## Step1: Property of rotation
Rotation is a rigid transformation. Rigid transformations preserve the length of segments.
## Step2: Analyze each option
Since $ \overline{ER} $ is rotated (a rigid transformation), $ E'R'=ER $. Option (1) $ E'R' = 2ER $ is wrong because rotation does not double the length. Option (2) $ E'R'=\frac{1}{2}ER $ is wrong as rotation does not halve  the length. Option (4) $ E'R'=ER + 30 $ is wrong because rotation does not add a constant to the length.

# Answer:
(3)

### Question 2
# Explanation:
## Step1: Property of rotation
In a rotation about a point $ C $, the distance from the center of rotation $ C $ to a pre - image point (e.g., $ H $) is equal to the distance from $ C $ to its image point (e.g., $ H' $), and  the distance from $ C $ to a pre - image point (e.g., $ I $) is equal to the distance from $ C $ to its image point (e.g., $ I' $). Also, the length of a segment $ HI $ is equal to the length of its image segment $ H'I' $.
## Step2: Analyze each option
Option (1): The distance from $ C $ to $ I $ (pre - image) and the distance from $ C $ to $ H' $ (image of a different point) do not have to be the same. Option (2): The distance from $ C $ to $ H $ (pre - image) and the distance from $ C $ to $ H' $ (image) are equal (property of rotation). Option (3): The distance from $ H $ to $ I $ (pre - image segment) and the distance from $ H' $ to $ I' $ (image segment) are equal (property of rotation). Option (4): The distance from $ I $ to $ C $ (pre - image) and the distance from $ C $ to $ I' $ (image) are equal (property of rotation).

# Answer:
(1)

### Question 3
# Explanation:
## Step1: Property of rotation
Rotation is a rigid transformation. Rigid transformations preserve the shape and size of a figure, which means that the corresponding angles of the pre - image ($ \triangle MNP $) and the image ($ \triangle QRS $) are equal.
## Step2: Identify the corresponding angle
In $ \triangle MNP $, $ \angle P=51^{\circ}$, $ \angle M = 59^{\circ}$, $ \angle N=70^{\circ}$. Since $ \triangle QRS$ is the image of $ \triangle MNP$ after rotation, the angles of $ \triangle QRS$ are equal to the angles of $ \triangle MNP$. So $ \angle R$ corresponds to $ \angle P$ (depending on the rotation mapping, but because of congruence of triangles under rotation) and $ \angle R = 51^{\circ}$

# Answer:
(1)

### Question 4
# Explanation:
## Step1: Rotation rules
The rule for a $90^{\circ}$ clockwise rotation about the origin $(x,y)\to(y, - x)$.
## Step2: Analyze each option
Option (1): $(x,y)\to(y,x)$ is a reflection over the line $y = x$. Option (2): $(x,y)\to(-y,x)$ is a $90^{\circ}$ counter - clockwise rotation about the origin. Option (3): $(x,y)\to(-x,-y)$ is a $180^{\circ}$ rotation about the origin.

# Answer:
(4)

### Question 5
# Explanation:
## Step1: Assume coordinates
Let's assume $W=(1,2)$ and $X=(4,8)$ (by looking at the grid - like structure in the problem, although not explicitly given in a coordinate - labeled way, we use the general rotation rule). The rule for a $90^{\circ}$ counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$.
## Step2: Apply the rule
For point $W=(1,2)$, after $90^{\circ}$ counter - clockwise rotation: $x = 1,y = 2$, the new coordinates $W'=(-2,1)$. For point $X=(4,8)$, after $90^{\circ}$ counter - clockwise rotation: $x = 4,y = 8$, the new coordinates $X'=(-8,4)$ (but if we assume a different coordinate assignment based on the visual (assuming $W=( - 1,2)$ and $X=( - 4,8)$ in a non - standard coordinate reading which is wrong, or using the formula correctly). Using the formula $(x,y)\to(-y,x)$ correctly:
If we assume $W=(1,2)$ (taking $x = 1,y=2$) and $X=(4,8)$ (taking $x = 4,y = 8$)
$W'=(-2,1)$ and $X'=(-8,4)$ (not in the options). But if we consider the formula application again, maybe a mis - read of coordinates. If $W=(2,1)$ ( $x = 2,y = 1$) and $X=(8,4)$ ( $x=8,y = 4$), after $90^{\circ}$ counter - clockwise rotation $(x,y)\to(-y,x)$, $W'=(-1,2)$ and $X'=(-4,8)$ (wrong). If we use the formula $(x,y)\to(-y,x)$ for $W=(1,2)$ gives $W'=(-2,1)$ and for $X=(4,8)$ gives $X'=(-8,4)$ (not in options). But if we consider the formula for $90^{\circ}$ counter - clockwise $(x,y)\to(-y,x)$ and assume the original points (by visual, if $W=(1,2)$ is mis - represented as $W=(2,1)$ in coordinate reading (wrong coordinate reading), no. Using the formula correctly:
Let’s use the general formula. For a point $(x,y)$ rotated $90^{\circ}$ counter - clockwise about the origin, the image is $(-y,x)$.
If we assume $W=(1,2)$ ( $x = 1,y=2$) $\to W'=(-2,1)$ and $X=(4,8)$ ( $x = 4,y = 8$)$\to X'=(-8,4)$ (not in options). But if we consider the options:
Option (4): $W'(-2,1)$ and $X'(8,4)$ (if there is a mis - take in the problem's option (maybe a typo in $X'$ sign), but using the formula $(x,y)\to(-y,x)$

# Answer:
(4)

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Property of rotation

Step2: Analyze each option

Step1: Property of rotation

Step2: Analyze each option

Step1: Property of rotation

Step2: Identify the corresponding angle

Answer:

Question 2