Question

What are the coordinates of a point symmetric to the origin with the point \((3,-9)\) ?

Short answer

The coordinates of the symmetric point are (-3, 9).

Step-by-step solution

Understanding Symmetry with Respect to the Origin

A point symmetric to another point with respect to the origin is a reflection of that point through the origin. To find its coordinates, both the x-coordinate and the y-coordinate change their signs.

Identify Original Coordinates

The given point is \((3, -9)\). Here, the x-coordinate is 3 and the y-coordinate is -9.

Change Signs of Coordinates

To find the symmetric point, multiply the x-coordinate and y-coordinate by -1. Thus, the new coordinates will be \(-3\) and \(+9\).

Write the Symmetric Point

After changing the signs, the coordinates of the symmetric point are \((-3, 9)\).

Key concepts

Symmetry with Respect to Origin

Symmetry with respect to the origin in coordinate geometry means that for each point \((x, y)\), there is a corresponding point \((-x, -y)\). This is like looking at a reflection of the point through the origin. The process involves reversing the signs of both the x-coordinate and the y-coordinate of the original point.
For example, if we have the point \((3, -9)\), its symmetric counterpart through the origin would be \((-3, 9)\). This concept is useful in problems that involve reflections or coordinate transformations.

Reflection of Points

The reflection of a point through the origin is akin to flipping the point over to the opposite quadrant.
If the original point is in the first quadrant, its reflection will be in the third quadrant, and vice versa.

• A point \((x, y)\) in the first quadrant would reflect to \((-x, -y)\) in the third quadrant.
• A point \((-x, y)\) in the second quadrant would reflect to \((x, -y)\) in the fourth quadrant.

Reflecting a point is straightforward: just change the signs of both coordinates.

Coordinate Transformation

Coordinate transformation is a common concept that involves changing the coordinates of a point in a plane.
To transform coordinates with respect to the origin, especially in the context of symmetry, you take the original coordinates \((x, y)\) and convert them to \((-x, -y)\). This process effectively moves the point to a new position in the coordinate system where it maintains a symmetric relationship with the origin.
Understanding coordinate transformation can help in solving more complex problems in geometry and physics.

Negative Coordinates

Negative coordinates indicate a position in the either the second, third, or fourth quadrants of the coordinate plane.
For example:

• A point \((x, -y)\) falls in the fourth quadrant.
• A point \((-x, y)\) falls in the second quadrant.
• A point \((-x, -y)\) falls in the third quadrant.

Negative coordinates can result from transformations that involve reflections or rotations around the origin.

Positive Coordinates

Positive coordinates pinpoint a position in the first or third quadrants of the coordinate plane. Specifically:

• A point \((x, y)\) is in the first quadrant, both coordinates are positive.
• After reflection through the origin, a point \((-x, -y)\) will also have both coordinates negative, landing in the third quadrant.

Positive coordinates provide a reference for understanding their negative counterparts in transformations and reflections.