IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Get the information you need from our community of experts, who provide detailed and trustworthy answers.

Select the correct answer.

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 :

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.