As an expert in the field of geometry and transformations, I can provide a comprehensive explanation of what happens when a point is rotated 90 degrees clockwise about the origin in a two-dimensional coordinate system.

When we talk about rotating a point around the origin, we are essentially moving the point around a circle centered at the origin (0, 0). The radius of this circle is the distance from the origin to the point being rotated. The rotation is a rigid motion, meaning it preserves distances and angles. A 90-degree clockwise rotation is a quarter turn to the right when looking down from above the plane.

Let's break down the process step by step:


1. Identifying the Point to Rotate: The first step is to identify the point that needs to be rotated. Let's use the point M (h, k) as an example, where h is the x-coordinate and k is the y-coordinate.


2. Understanding the Rotation: A 90-degree clockwise rotation about the origin can be visualized as follows: if you place your right hand on the paper with your thumb pointing in the direction of the positive x-axis and your fingers curled towards the positive y-axis, a clockwise rotation will be in the direction your fingers curl. This is a standard way to visualize rotations in mathematics.


3. Applying the Rotation: To apply a 90-degree clockwise rotation to point M (h, k), we need to find the new coordinates of the point after the rotation. The transformation rules for a 90-degree clockwise rotation about the origin are as follows:
- The new x-coordinate (x') of the point will be the negative of the original y-coordinate (k).
- The new y-coordinate (y') of the point will be the original x-coordinate (h).

This gives us the new point M' (k, -h).


4. Worked-out Examples: Let's apply this to the example provided:
- Given point M (-2, 3), we want to find the new position M' after a 90-degree clockwise rotation about the origin.
- Using the transformation rules, the new x-coordinate will be -3 (the negative of the original y-coordinate), and the new y-coordinate will be 2 (the original x-coordinate).
- Therefore, the new position of point M after the rotation will be M' (3, -2).


5. Graphical Representation: If you were to plot this on graph paper, you would start by plotting point M (-2, 3). Then, to find M', you would move three units up (since the y-coordinate is positive) and two units to the right (since the x-coordinate is negative after the rotation). The resulting point would be M' (3, -2), which is the reflection of M across the y-axis.


6. Implications of the Rotation: It's important to note that a 90-degree clockwise rotation is equivalent to a reflection across the y-axis followed by a 180-degree rotation about the y-axis. This means that the x-coordinates of points will become their negatives, and the y-coordinates will remain the same but will be reflected across the y-axis.

7.
Generalizing the Concept: The concept of rotating a point can be generalized to any angle of rotation. For angles other than 90 degrees, the transformation would involve using trigonometric functions to find the new coordinates. However, for a 90-degree rotation, the process is straightforward and does not require trigonometry.

In conclusion, rotating a point 90 degrees clockwise about the origin is a simple yet powerful concept in geometry. It allows us to understand how shapes and figures can be transformed in the plane, which has applications in fields ranging from art to engineering. Understanding these transformations is fundamental to grasping more complex geometric concepts and spatial reasoning.

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The new position of point M (h, k) will become M' (k, -h). Worked-out examples on 90 degree clockwise rotation about the origin: 1. Plot the point M (-2, 3) on the graph paper and rotate it through 90-- in clockwise direction, about the origin.read more >>