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Finding the Coordinate for a [tex]$180^{\circ}$[/tex] Rotation

Point A has coordinates [tex]$A (3,2)$[/tex]. The point is rotated [tex]$180^{\circ}$[/tex] clockwise around the origin.

Find the coordinates of point [tex]$A^{\prime}$[/tex].

A. [tex]$A^{\prime}(2,3)$[/tex]
B. [tex]$A^{\prime}(-3,-2)$[/tex]
C. [tex]$A^{\prime}(-2,-3)$[/tex]
D. [tex]$A^{\prime}(-2,3)$[/tex]



Answer :

GinnyAnswer
To find the coordinates of point [tex]\( A \)[/tex] after a [tex]\( 180^\circ \)[/tex] clockwise rotation around the origin, we can follow these steps:

1. Understand the Effect of a [tex]\( 180^\circ \)[/tex] Rotation:
Rotating a point [tex]\( (x, y) \)[/tex] by [tex]\( 180^\circ \)[/tex] clockwise around the origin results in a point that has its coordinates negated. This means:
[tex]\[ A'(x', y') = (-x, -y) \][/tex]

2. Identify the Original Coordinates:
The original coordinates of point [tex]\( A \)[/tex] are given as:
[tex]\[ A (3, 2) \][/tex]

3. Negate Both Coordinates:
To find the new coordinates [tex]\( A' \)[/tex] after the rotation, we negate both the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( A \)[/tex]:
[tex]\[ A'_x = -A_x = -3 \][/tex]
[tex]\[ A'_y = -A_y = -2 \][/tex]

4. Determine the New Coordinates:
Thus, the new coordinates of point [tex]\( A' \)[/tex] after the [tex]\( 180^\circ \)[/tex] clockwise rotation are:
[tex]\[ A'(-3, -2) \][/tex]

Given the results, the correct coordinate for point [tex]\( A' \)[/tex] after a [tex]\( 180^\circ \)[/tex] rotation is:
[tex]\[ A'(-3, -2) \][/tex]

So, the correct option is:
[tex]\[ A'(-3, -2) \][/tex]

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