To determine which set of ordered pairs exhibits point symmetry with respect to the origin [tex]\((0,0)\)[/tex], we need to understand that point symmetry about the origin means if [tex]\((x, y)\)[/tex] is a point on the shape, then [tex]\((-x, -y)\)[/tex] must also be a point on the shape.
Let's examine each option:
### Option 1: [tex]\((5,4)\)[/tex], [tex]\((-5,-4)\)[/tex]
- The first point is [tex]\((5,4)\)[/tex].
- The second point is [tex]\((-5,-4)\)[/tex].
To check point symmetry about the origin:
- The coordinates of the second point are the negatives of the coordinates of the first point:
- [tex]\(x_2 = -5 = -x_1\)[/tex]
- [tex]\(y_2 = -4 = -y_1\)[/tex]
This shows that [tex]\((5,4)\)[/tex] and [tex]\((-5,-4)\)[/tex] are symmetric with respect to the origin.
### Option 2: [tex]\((5,4)\)[/tex], [tex]\((5,-4)\)[/tex]
- The first point is [tex]\((5,4)\)[/tex].
- The second point is [tex]\((5,-4)\)[/tex].
To check point symmetry about the origin:
- For point symmetry, the second point should be [tex]\((-x_1, -y_1) = (-5, -4)\)[/tex].
However, this second point is [tex]\((5, -4)\)[/tex], not [tex]\((-5,-4)\)[/tex]. Therefore, they are not symmetric with respect to the origin.
### Option 3: [tex]\((5,4)\)[/tex], [tex]\((4,5)\)[/tex]
- The first point is [tex]\((5,4)\)[/tex].
- The second point is [tex]\((4,5)\)[/tex].
To check point symmetry about the origin:
- For point symmetry, the second point should be [tex]\((-x_1, -y_1) = (-5, -4)\)[/tex].
However, this second point is [tex]\((4, 5)\)[/tex], not [tex]\((-5,-4)\)[/tex]. Therefore, they are not symmetric with respect to the origin.
### Option 4: [tex]\((5,4)\)[/tex], [tex]\((-5,4)\)[/tex]
- The first point is [tex]\((5,4)\)[/tex].
- The second point is [tex]\((-5,4)\)[/tex].
To check point symmetry about the origin:
- For point symmetry, the second point should be [tex]\((-x_1, -y_1) = (-5, -4)\)[/tex].
However, this second point is [tex]\((-5, 4)\)[/tex], not [tex]\((-5,-4)\)[/tex]. Therefore, they are not symmetric with respect to the origin.
Based on our analysis, the set of points [tex]\((5,4)\)[/tex] and [tex]\((-5,-4)\)[/tex] has point symmetry with respect to the origin [tex]\((0,0)\)[/tex].
Therefore, the correct answer is:
- [tex]\((5,4)\)[/tex], [tex]\((-5,-4)\)[/tex]
