question 39 of 39 which transformations can be used to carry abcd onto itself? the point of rotation is (3,2). check all that apply. a. translation two units down b. reflection across the line x = 3 c. reflection across the line y = 2 d. rotation of 90°

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Accepted Answer

# Answer:
B. Reflection across the line $x = 3$, C. Reflection across the line $y = 2$

# Explanation:
## Step1: Analyze reflection across $x = 3$
The line $x=3$ is the vertical line passing through the mid - point of $AB$ and $CD$. For a point $(x,y)$ in the rectangle $ABCD$, the reflection of a point $(x,y)$ across the line $x = 3$ is $(6 - x,y)$. If we take point $A(1,1)$, its reflection across $x = 3$ is $(5,1)$ (which is point $B$), point $B(5,1)$ reflects to $(1,1)$ (point $A$), point $C(5,3)$ reflects to $(1,3)$ (point $D$) and point $D(1,3)$ reflects to $(5,3)$ (point $C$). So reflection across $x = 3$ maps $ABCD$ onto itself.
## Step2: Analyze reflection across $y = 2$
The line $y = 2$ is the horizontal line passing through the mid - point of $AD$ and $BC$. For a point $(x,y)$ in the rectangle $ABCD$, the reflection of a point $(x,y)$ across the line $y=2$ is $(x,4 - y)$. If we take point $A(1,1)$, its reflection across $y = 2$ is $(1,3)$ (point $D$), point $D(1,3)$ reflects to $(1,1)$ (point $A$), point $B(5,1)$ reflects to $(5,3)$ (point $C$) and point $C(5,3)$ reflects to $(5,1)$ (point $B$). So reflection across $y = 2$ maps $ABCD$ onto itself.
## Step3: Analyze translation two units down
If we translate the rectangle two units down, the $y$ - coordinate of each point $(x,y)$ becomes $(x,y - 2)$. Point $A(1,1)$ will be $(1,-1)$ which is not on the original rectangle $ABCD$. So this translation does not map $ABCD$ onto itself.
## Step4: Analyze rotation of $90^{\circ}$
A $90^{\circ}$ rotation about the point $(3,2)$:
The rotation formula about a point $(h,k)$ is $\begin{cases}x'=(x - h)\cos\theta-(y - k)\sin\theta+h\y'=(x - h)\sin\theta+(y - k)\cos\theta + k\end{cases}$ with $\theta = 90^{\circ}$, $\cos\theta=0$, $\sin\theta = 1$.
For point $A(1,1)$: $x'=(1 - 3)\times0-(1 - 2)\times1+3=4$, $y'=(1 - 3)\times1+(1 - 2)\times0+2=0$. The point $(4,0)$ is not on the original rectangle $ABCD$. So a $90^{\circ}$ rotation about $(3,2)$ does not map $ABCD$ onto itself.

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Question

Explanation:

Step1: Analyze reflection across \(x = 3\)

Step2: Analyze reflection across \(y = 2\)

Step3: Analyze translation two units down

Step4: Analyze rotation of \(90^{\circ}\)

Answer: