Q: Final Rotation Formulas:

###\begin{aligned} x' &= x \cos\theta - y \sin\theta \\ y' &= x \sin\theta + y \cos\theta \end{aligned}###

Q: Illustrative Examples of Rotations

Example 1: Rotate the point B(2, 2) by 45 degrees counterclockwise around the origin.

Q: Illustrative Examples of Reflections

Example 1: Reflect the point G(-1, -4) across the x-axis.

Question 1

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Understanding Rotations and Reflections
In coordinate geometry, transformations involve moving points and shapes in the plane.
Rotations involve turning a figure around a fixed point (the center of rotation) by a certain angle. Think of it like spinning a wheel; every point on the wheel moves in a circular path around the center. The angle of rotation determines how far the figure is turned. Rotations preserve the size and shape of the figure; they are isometries.
Reflections, on the other hand, involve “flipping” a figure over a line (the line of reflection). Imagine a mirror; the reflection of a point is the same distance from the mirror line as the original point, but on the opposite side. Reflections also preserve the size and shape of the figure (making them isometries), but they change the orientation. For example, a clockwise rotation becomes a counterclockwise rotation after reflection.
Rotations: Turning Points and Shapes
Rotations are defined by a center of rotation and an angle of rotation. The most common center of rotation is the origin (0, 0). We focus on rotations around the origin for simplicity. The angle of rotation is usually measured in degrees or radians, with positive angles indicating counterclockwise rotations and negative angles indicating clockwise rotations. The key to understanding rotations lies in how the coordinates of a point change.
Rotations Around the Origin

Accepted Answer

To rotate a point ##(x, y)## counterclockwise by an angle ##	heta## around the origin, we use the following formulas. These formulas are derived using trigonometry and the properties of circles. The new coordinates ##(x', y')## are given by:

Question 2

Final Rotation Formulas:

Accepted Answer

###\begin{aligned} x' &= x \cos	heta - y \sin	heta \ y' &= x \sin	heta + y \cos	heta \end{aligned}###

Question 3

Illustrative Examples of Rotations

Accepted Answer

Example 1: Rotate the point B(2, 2) by 45 degrees counterclockwise around the origin.

Question 4

Reflections Across the x-axis

Accepted Answer

When reflecting a point across the x-axis, the x-coordinate remains the same, and the y-coordinate changes sign. The rule is:

Question 5

Reflections Across the y-axis

Accepted Answer

When reflecting a point across the y-axis, the y-coordinate remains the same, and the x-coordinate changes sign. The rule is:

Question 6

Reflections Across the line y = x

Accepted Answer

When reflecting a point across the line y = x, the x and y coordinates are swapped. The rule is:

Question 7

Illustrative Examples of Reflections

Accepted Answer

Example 1: Reflect the point G(-1, -4) across the x-axis.

Question 8

Example: Combining Rotation and Reflection

Accepted Answer

Let’s consider a point J(1, 1). First, we rotate it 90 degrees counterclockwise around the origin. Using the rotation formulas:

Question 9

Problem 1:

Accepted Answer

Rotate the point (3, 4) by 90 degrees clockwise around the origin.

Question 10

Problem 2:

Accepted Answer

Reflect the point (-2, 5) across the line y = x.

Question 11

Problem 3:

Accepted Answer

Reflect the point (1, -1) across the x-axis.

Question 12

Problem 4:

Accepted Answer

Rotate the point (0, 5) by 180 degrees around the origin.

Question 13

Problem 5:

Accepted Answer

Reflect the point (3, 0) across the y-axis.

SHORT NOTES

Rotations and Reflections in Coordinate Geometry

Radians

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