1. suppose $ \triangle rst $ is translated and then reflected to form $ \triangle r ^ { prime } s ^ { prime } t ^ { prime } $. select all of the statements that are true.
a. length $ overline { r ^ { prime } s ^ { prime } } = 4.3 \mathrm { cm } $
b. $ \angle r \cong \angle r ^ { prime } $
c. $ m \angle t ^ { prime } = 71 $
d. $ overline { s t } \cong overline { t ^ { prime } s ^ { prime } } $
e. length $ overline { r t } $ length $ overline { r ^ { prime } t ^ { prime } } $
2. point $ t $ is at $ ( - 2,5 ) $. what are the coordinates of point $ t ^ { prime } $ after a reflection across $ x = 0 $ and then $ y = 0 ? ( , ) $
3. $ overline { p q } $ is reflected across the line $ x = - 1 $. select all that apply.
a. $ overline { p q } \cong overline { p ^ { prime } q ^ { prime } } $
b. $ q ^ { prime } $ is plotted at $ ( - 4,4 ) $.
c. $ q ^ { prime } $ is plotted at $ ( - 4 , - 4 ) $.
d. $ q $ and $ q ^ { prime } $ are in the same quadrant.
e. $ p ^ { prime } $ and $ p $ are in the same quadrant.
4. which transformation or sequence of transformations maps $ \triangle a b c $ to $ \triangle a ^ { prime } b ^ { prime } c ^ { prime } $ ?
(a) a translation 5 units left
(b) a reflection across $ x = 0 $, then a translation 5 units left
(c) a reflection across $ y = 0 $, then a translation 5 units left
(d) a rotation $ 90 ^ { \circ } $ about the origin
5. point $ p ^ { prime } ( 5 , - 4 ) $ is the image of point $ p ( 2,3 ) $ under a translation. select the image of $ ( 6 , - 2 ) $ under the same translation.
(a) $ ( 7 , - 1 ) $
(b) $ ( 13 , - 3 ) $
(c) $ ( 9 , - 9 ) $
(d) $ ( 3,5 ) $
6. given point $ a $ is not on line $ m $, suppose $ a $ is reflected across line $ m $. which of the following is true?
(a) $ overline { a a ^ { prime } } $ is the bisector of line $ m $.
(b) line $ m $ is the perpendicular bisector of $ overline { a a ^ { prime } } $.
(c) line $ m $ is parallel to $ overline { a a ^ { prime } } $.
(d) $ a = a ^ { prime } $

Question

1. suppose $ \triangle rst $ is translated and then reflected to form $ \triangle r ^ { prime } s ^ { prime } t ^ { prime } $. select all of the statements that are true.
a. length $ overline { r ^ { prime } s ^ { prime } } = 4.3 \mathrm { cm } $
b. $ \angle r \cong \angle r ^ { prime } $
c. $ m \angle t ^ { prime } = 71 $
d. $ overline { s t } \cong overline { t ^ { prime } s ^ { prime } } $
e. length $ overline { r t } > $ length $ overline { r ^ { prime } t ^ { prime } } $
2. point $ t $ is at $ ( - 2,5 ) $. what are the coordinates of point $ t ^ { prime } $ after a reflection across $ x = 0 $ and then $ y = 0 ? ( , ) $
3. $ overline { p q } $ is reflected across the line $ x = - 1 $. select all that apply.
a. $ overline { p q } \cong overline { p ^ { prime } q ^ { prime } } $
b. $ q ^ { prime } $ is plotted at $ ( - 4,4 ) $.
c. $ q ^ { prime } $ is plotted at $ ( - 4 , - 4 ) $.
d. $ q $ and $ q ^ { prime } $ are in the same quadrant.
e. $ p ^ { prime } $ and $ p $ are in the same quadrant.
4. which transformation or sequence of transformations maps $ \triangle a b c $ to $ \triangle a ^ { prime } b ^ { prime } c ^ { prime } $ ?
(a) a translation 5 units left
(b) a reflection across $ x = 0 $, then a translation 5 units left
(c) a reflection across $ y = 0 $, then a translation 5 units left
(d) a rotation $ 90 ^ { \circ } $ about the origin
5. point $ p ^ { prime } ( 5 , - 4 ) $ is the image of point $ p ( 2,3 ) $ under a translation. select the image of $ ( 6 , - 2 ) $ under the same translation.
(a) $ ( 7 , - 1 ) $
(b) $ ( 13 , - 3 ) $
(c) $ ( 9 , - 9 ) $
(d) $ ( 3,5 ) $
6. given point $ a $ is not on line $ m $, suppose $ a $ is reflected across line $ m $. which of the following is true?
(a) $ overline { a a ^ { prime } } $ is the bisector of line $ m $.
(b) line $ m $ is the perpendicular bisector of $ overline { a a ^ { prime } } $.
(c) line $ m $ is parallel to $ overline { a a ^ { prime } } $.
(d) $ a = a ^ { prime } $

Accepted Answer

# Explanation:
## Step1: Analyze problem 1
- **Translation and reflection**: These are rigid transformations. Rigid transformations preserve side - lengths and angle - measures.
- **For option A**: Since $ST = 4.3\mathrm{cm}$ and rigid transformations preserve side - lengths, $R'S'=ST = 4.3\mathrm{cm}$.
- **For option B**: $\angle R$ and $\angle R'$ are corresponding angles. Rigid transformations preserve angle - measures, so $\angle R\cong\angle R'$.
- **For option C**: $m\angle T=m\angle T' = 71^{\circ}$ because rigid transformations preserve angle - measures.
- **For option D**: $\overline{ST}\cong\overline{S'T'}$, not $\overline{ST}\cong\overline{T'S'}$.
- **For option E**: Since rigid transformations preserve side - lengths, $RT = R'T'$.

## Step2: Analyze problem 2
- **Reflection across $x = 0$ (y - axis)**: The rule for reflection across the $y$ - axis is $(x,y)\to(-x,y)$. For the point $T(-2,5)$, after reflection across $x = 0$, the point becomes $(2,5)$.
- **Reflection across $y = 0$ (x - axis)**: The rule for reflection across the $x$ - axis is $(x,y)\to(x, - y)$. After reflection of $(2,5)$ across $y = 0$, the point $T'$ is $(2,-5)$.

## Step3: Analyze problem 3
- **Reflection across $x=-1$**:
    - **For option A**: Reflection is a rigid transformation, so $\overline{PQ}\cong\overline{P'Q'}$.
    - **For option B and C**: The distance between a point $(x,y)$ and the line $x = a$ is $|x - a|$. For point $Q(2,4)$, the distance from $x=-1$ is $|2-(-1)|=3$. The $x$ - coordinate of $Q'$ is $-1-3=-4$, and the $y$ - coordinate remains the same. So $Q'(-4,4)$.
    - **For option D**: $Q(2,4)$ is in the first quadrant and $Q'(-4,4)$ is in the second quadrant.
    - **For option E**: $P(-1,-3)$ (assuming from the graph), after reflection across $x=-1$, $P'(-1,-3)$ (since $P$ is on the line $x = - 1$), so $P$ and $P'$ are in the same quadrant.

## Step4: Analyze problem 4
- **For option A**: A translation 5 units left:
    - For point $A(5,5)$, $A'(0,5)
eq A'(0, - 5)$.
- **For option B**:
    - **Reflection across $x = 0$ (y - axis)**: The rule is $(x,y)\to(-x,y)$. For $A(5,5)\to(-5,5)$, $B(1,3)\to(-1,3)$, $C(2,1)\to(-2,1)$.
    - **Translation 5 units left**: The rule is $(x,y)\to(x - 5,y)$. For $(-5,5)\to(0,5)
eq(0,-5)$.
- **For option C**:
    - **Reflection across $y = 0$ (x - axis)**: The rule is $(x,y)\to(x,-y)$. For $A(5,5)\to(5,-5)$, $B(1,3)\to(1,-3)$, $C(2,1)\to(2,-1)$.
    - **Translation 5 units left**: The rule is $(x,y)\to(x - 5,y)$. For $(5,-5)\to(0,-5)$, $(1,-3)\to(-4,-3)$, $(2,-1)\to(-3,-1)$.
- **For option D**:
    - **Rotation $90^{\circ}$ about the origin**: The rule for a $90^{\circ}$ clockwise rotation about the origin is $(x,y)\to(y,-x)$. For $A(5,5)\to(5,-5)
eq(0,-5)$.

## Step5: Analyze problem 5
- **Find the translation vector**:
    - If $P(2,3)$ is translated to $P'(5,-4)$, the translation vector $(a,b)$ is found by $(x + a,y + b)$. So $a=5 - 2=3$ and $b=-4 - 3=-7$.
    - **Apply the translation to $(6,-2)$**: Using the rule $(x,y)\to(x + 3,y-7)$, we get $(6 + 3,-2-7)=(9,-9)$.

## Step6: Analyze problem 6
- **Reflection properties**:
    - When a point $A$ is reflected across a line $m$ (where $A
otin m$), the line $m$ is the perpendicular bisector of the segment $\overline{AA'}$.
    - **For option A**: $\overline{AA'}$ is perpendicular to line $m$ (not a bisector of $m$).
    - **For option C**: Line $m$ is perpendicular to $\overline{AA'}$, not parallel.
    - **For option D**: $A
eq A'$ since $A$ is not on line $m$.

# Answer:
1. A, B, C
2. $(2,-5)$
3. A, B, E
4. C
5. C
6. B

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Analyze problem 1

Step2: Analyze problem 2

Step3: Analyze problem 3

Step4: Analyze problem 4

Step5: Analyze problem 5

Step6: Analyze problem 6

Answer: