To solve the problem, let's approach it step-by-step.
### Step 1: Rotate each point [tex]\(180^\circ\)[/tex] about the origin.
Rotating a point [tex]\( (x, y) \)[/tex] by [tex]\(180^\circ\)[/tex] about the origin results in the new coordinates [tex]\( (-x, -y) \)[/tex].
#### Rotating Points:
1. Point [tex]\(A(1, 1)\)[/tex]:
[tex]\[A' = (-1, -1)\][/tex]
2. Point [tex]\(B(5, 4)\)[/tex]:
[tex]\[B' = (-5, -4)\][/tex]
3. Point [tex]\(C(7, 1)\)[/tex]:
[tex]\[C' = (-7, -1)\][/tex]
4. Point [tex]\(D(3, -2)\)[/tex]:
[tex]\[D' = (-3, 2)\][/tex]
### Step 2: Translate each point 5 units to the right and 1 unit down.
The translation involves adding 5 to the x-coordinate and subtracting 1 from the y-coordinate.
#### Translating Points:
1. Point [tex]\(A' = (-1, -1)\)[/tex]:
[tex]\[ A'' = ( -1 + 5, -1 - 1) = (4, -2) \][/tex]
2. Point [tex]\(B' = (-5, -4)\)[/tex]:
[tex]\[ B'' = ( -5 + 5, -4 - 1) = (0, -5) \][/tex]
3. Point [tex]\(C' = (-7, -1)\)[/tex]:
[tex]\[ C'' = ( -7 + 5, -1 - 1) = (-2, -2) \][/tex]
4. Point [tex]\(D' = (-3, 2)\)[/tex]:
[tex]\[ D'' = ( -3 + 5, 2 - 1) = (2, 1) \][/tex]
### Step 3: Compile the new coordinates into the transformed parallelogram:
[tex]\[
A''(4, -2), B''(0, -5), C''(-2, -2), D''(2, 1)
\][/tex]
### Step 4: Match the coordinates with the given choices:
The correct set of transformed coordinates matches:
B. [tex]\( A''(4, -2), B''(0, -5), C''(-2, -2), D''(2, 1) \)[/tex]
Thus, the correct answer is B.
