Problem 1 - Algebra and Trigonometry

Question

If the rectangular coordinates of a point are (4,-6) the point symmetric to it with respect to the origin is ________.

Step-by-Step Solution

Answer

(-4, 6)

Step 1: Identify the coordinates of the given point

The given point in rectangular coordinates is (4, -6). These are the coordinates (x, y) of the point.

Step 2: Understand the concept of symmetry with respect to the origin

A point symmetric to another point with respect to the origin is found by negating both the x-coordinate and the y-coordinate of the original point.

Step 3: Calculate the symmetric point

Negate the x-coordinate: -4 and negate the y-coordinate: 6. Thus, the coordinates of the symmetric point are (-4, 6).

Step 4: Write the coordinates of the symmetric point

The point symmetric to (4, -6) with respect to the origin is (-4, 6).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent the location of a point in a 2-dimensional plane using ordered pairs (x, y). The x-coordinate specifies the horizontal position, while the y-coordinate specifies the vertical position. The origin, located at (0,0), is the central reference point where the x-axis and y-axis intersect. For example, the point (4, -6) means that you move 4 units to the right along the x-axis, and then 6 units down along the y-axis. This notation is extremely useful in graphing and understanding mathematical relationships in a visual format.

Point Symmetry

Point symmetry refers to a situation where a point is reflected or mirrored through another central point, resulting in a symmetric image. One common central point for symmetry is the origin. If you have a point (x, y) and you want to find its symmetric partner with respect to a certain point, the coordinates of the new point will be a reflection across the central point. For origin symmetry, you simply negate both the x and y coordinates of the original point, turning (x, y) into (-x, -y). This process transforms the point while maintaining its distance from the origin, but in the opposite direction.

Origin Symmetry

Origin symmetry is a specific type of point symmetry where the central point of reflection is the origin (0,0). If you have a point with coordinates (4, -6), the symmetric point with respect to the origin is found by negating both coordinates.

Here's how it works:

• Negate the x-coordinate: from 4 to -4
• Negate the y-coordinate: from -6 to 6

Thus, the coordinates of the symmetric point are (-4, 6). This means if you start at the origin, you would move 4 units to the left and 6 units up to reach the new point, essentially flipping the original point to the opposite quadrant.

Negating Coordinates

Negating coordinates is the mathematical process used to achieve origin symmetry. To negate a coordinate means to change its sign: positive to negative or negative to positive. This simple action has a profound effect by relocating the point to a different quadrant in the coordinate system.

For example:

• If x = 4, negating it gives -4
• If y = -6, negating it gives 6

After negation, the point (4, -6) is transformed to (-4, 6). This technique can be useful in various areas of mathematics, including reflections, rotations, and translations in geometry.