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A parallelogram has coordinates [tex]\( A(1,1), B(5,4), C(7,1), \)[/tex] and [tex]\( D(3,-2) \)[/tex]. What are the coordinates of parallelogram [tex]\( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \)[/tex] after a [tex]\( 180^{\circ} \)[/tex] rotation about the origin and a translation 5 units to the right and 1 unit down?

A. [tex]\( A^{\prime}(-4,-2), B^{\prime}(0,-5), C^{\prime}(2,-2), D^{\prime}(-2,1) \)[/tex]

B. [tex]\( A^{\prime}(4,-2), B^{\prime}(0,-5), C^{\prime}(-2,-2), D^{\prime}(2,1) \)[/tex]

C. [tex]\( A^{\prime}(4,2), B^{\prime}(0,5), C^{\prime}(-2,2), D^{\prime}(2,-1) \)[/tex]

D. [tex]\( A^{\prime}(-4,2), B^{\prime}(0,5), C^{\prime}(2,2), D^{\prime}(-2,-1) \)[/tex]

Sagot :

GinnyAnsweridn
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.
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