To solve this problem, we need to find three different symmetric points relative to the given point [tex]\((7, -2)\)[/tex] based on the specified axes.
### 1. Symmetry with respect to the x-axis
When finding the symmetric point with respect to the x-axis, we reflect the point over the x-axis. This involves changing the sign of the y-coordinate while keeping the x-coordinate the same.
Given point: [tex]\((7, -2)\)[/tex]
To find the symmetric point:
- Keep the x-coordinate as 7
- Change the sign of the y-coordinate: [tex]\(-2\)[/tex] becomes [tex]\(2\)[/tex]
Thus, the symmetric point with respect to the x-axis is [tex]\((7, 2)\)[/tex].
### 2. Symmetry with respect to the y-axis
When finding the symmetric point with respect to the y-axis, we reflect the point over the y-axis. This involves changing the sign of the x-coordinate while keeping the y-coordinate the same.
Given point: [tex]\((7, -2)\)[/tex]
To find the symmetric point:
- Change the sign of the x-coordinate: [tex]\(7\)[/tex] becomes [tex]\(-7\)[/tex]
- Keep the y-coordinate as [tex]\(-2\)[/tex]
Thus, the symmetric point with respect to the y-axis is [tex]\((-7, -2)\)[/tex].
### 3. Symmetry with respect to the origin
When finding the symmetric point with respect to the origin, we reflect the point over the origin. This involves changing the signs of both the x-coordinate and the y-coordinate.
Given point: [tex]\((7, -2)\)[/tex]
To find the symmetric point:
- Change the sign of the x-coordinate: [tex]\(7\)[/tex] becomes [tex]\(-7\)[/tex]
- Change the sign of the y-coordinate: [tex]\(-2\)[/tex] becomes [tex]\(2\)[/tex]
Thus, the symmetric point with respect to the origin is [tex]\((-7, 2)\)[/tex].
### Summary
The symmetric points are:
- With respect to the x-axis: [tex]\((7, 2)\)[/tex]
- With respect to the y-axis: [tex]\((-7, -2)\)[/tex]
- With respect to the origin: [tex]\((-7, 2)\)[/tex]
These points provide the required reflections of the point [tex]\((7, -2)\)[/tex] relative to the specified axes.
