What is the ordered pair of [tex]$X^{\prime}[tex]$[/tex] after point [tex]$[/tex]X(3,4)[tex]$[/tex] is rotated [tex]$[/tex]180^{\circ}$[/tex]?

A. [tex]$X^{\prime}(3,-4)$[/tex]
B. [tex]$X^{\prime}(-3,-4)$[/tex]
C. [tex]$X^{\prime}(-4,3)$[/tex]
D. [tex]$X^{\prime}(-4,-3)$[/tex]



Answer :

To find the ordered pair of \( X' \) after rotating the point \( X(3, 4) \) by \( 180^\circ \) around the origin, let's follow these steps:

1. Identify the original coordinates of the point \( X \):
- The original coordinates are \( (3, 4) \).

2. Understand the effect of a \( 180^\circ \) rotation around the origin:
- Rotating a point \( (x, y) \) by \( 180^\circ \) around the origin transforms the coordinates to \( (-x, -y) \).

3. Apply the transformation:
- For the point \( X(3, 4) \):
- The new \( x \)-coordinate will be \( -3 \) (since \( x = 3 \) and \( -x = -3 \)).
- The new \( y \)-coordinate will be \( -4 \) (since \( y = 4 \) and \( -y = -4 \)).

4. Write the new ordered pair \( X' \):
- The ordered pair after rotation is \( (-3, -4) \).

Therefore, the ordered pair of [tex]\( X' \)[/tex] after rotating point [tex]\( X(3, 4) \)[/tex] by [tex]\( 180^\circ \)[/tex] is [tex]\( \boxed{(-3, -4)} \)[/tex].