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Question 9 of 36

Parallelogram [tex]$ABCD$[/tex] has vertex coordinates [tex]$A(0,1)$[/tex], [tex]$B(1,3)$[/tex], [tex]$C(4,3)$[/tex], and [tex]$D(3,1)$[/tex]. It is translated 1 unit to the right and 3 units down, and then rotated [tex]$180^{\circ}$[/tex] clockwise around the origin. What are the coordinates of [tex]$A^{\prime \prime}$[/tex]?

A. [tex]$(-2,-2)$[/tex]
B. [tex]$(-2,2)$[/tex]
C. [tex]$(-1,-3)$[/tex]
D. [tex]$(-1,2)$[/tex]



Answer :

GinnyAnswer
Sure! Let's walk through the problem step by step.

### Step 1: Initial Coordinates of Point A
We start with the coordinates of point [tex]\(A\)[/tex]:
[tex]\[ A(0, 1) \][/tex]

### Step 2: Translation
Next, we translate point [tex]\(A\)[/tex] 1 unit to the right and 3 units down.

- Moving 1 unit to the right means adding 1 to the x-coordinate:
[tex]\[ x_{\text{new}} = 0 + 1 = 1 \][/tex]

- Moving 3 units down means subtracting 3 from the y-coordinate:
[tex]\[ y_{\text{new}} = 1 - 3 = -2 \][/tex]

So the new coordinates after the translation are:
[tex]\[ A' = (1, -2) \][/tex]

### Step 3: Rotation
Now, we rotate the translated point [tex]\(A'\)[/tex] (i.e., [tex]\( (1, -2) \)[/tex]) [tex]\(180^{\circ}\)[/tex] clockwise about the origin.

A [tex]\(180^{\circ}\)[/tex] clockwise rotation around the origin will transform any point [tex]\((x, y)\)[/tex] into [tex]\((-x, -y)\)[/tex].

Applying this rule:
[tex]\[ x_{\text{rotated}} = -1 \][/tex]
[tex]\[ y_{\text{rotated}} = 2 \][/tex]

Thus, the final coordinates of point [tex]\(A''\)[/tex] after both the translation and rotation are:
[tex]\[ A'' = (-1, 2) \][/tex]

### Conclusion
The correct coordinates of [tex]\(A''\)[/tex] are:
[tex]\[ \boxed{(-1, 2)} \][/tex]

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