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The vertices of [tex]\(\triangle GHI\)[/tex] are [tex]\(G(-4,0), H(5,2)\)[/tex], and [tex]\(I(-2,-3)\)[/tex]. [tex]\(\triangle GHI\)[/tex] is rotated [tex]\(270^{\circ}\)[/tex] counterclockwise about the origin to form [tex]\(\triangle G'H'I'\)[/tex]. What are the coordinates of the vertices of [tex]\(\triangle G'H'I'\)[/tex]?

A. [tex]\(G'(-4,0), H'(5,2), I'(-2,-3)\)[/tex]

B. [tex]\(G'(0,4), H'(-2,-5), I'(3,2)\)[/tex]

C. [tex]\(G'(0,4), H'(2,-5), I'(-3,2)\)[/tex]

D. [tex]\(G'(4,0), H'(-5,2), I'(2,-3)\)[/tex]



Answer :

GinnyAnswer
To solve the problem of rotating the vertices [tex]\(G(-4,0)\)[/tex], [tex]\(H(5,2)\)[/tex], and [tex]\(I(-2,-3)\)[/tex] counterclockwise by [tex]\(270^\circ\)[/tex] about the origin, we need to transform each vertex using the properties of rotation.

### Step-by-Step Solution

Rotation matrix for [tex]\(270^\circ\)[/tex] counterclockwise:
When rotating a point [tex]\((x, y)\)[/tex] counterclockwise by [tex]\(270^\circ\)[/tex], the rotation transformation is equivalent to rotating [tex]\(90^\circ\)[/tex] clockwise. The transformation for [tex]\(90^\circ\)[/tex] clockwise is given by:
[tex]\[ (x', y') = (y, -x) \][/tex]

Using this formula:

1. Rotate [tex]\(G(-4, 0)\)[/tex]:
[tex]\[ G' = (0, -(-4)) = (0, 4) \][/tex]

2. Rotate [tex]\(H(5, 2)\)[/tex]:
[tex]\[ H' = (2, -5) \][/tex]

3. Rotate [tex]\(I(-2, -3)\)[/tex]:
[tex]\[ I' = (-3, 2) \][/tex]

### Result
Therefore, the coordinates of the vertices of [tex]\(\triangle G'H'I'\)[/tex] after a [tex]\(270^\circ\)[/tex] counterclockwise rotation are:
- [tex]\(G' = (0, 4)\)[/tex]
- [tex]\(H' = (2, -5)\)[/tex]
- [tex]\(I' = (-3, 2)\)[/tex]

### Answer
From the given multiple-choice options, the correct one is:
C. [tex]\(G'(0, 4), H'(2, -5), I'(-3, 2)\)[/tex]

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