Reflections and rotations

• Reflections and rotations are significant aspects of transformation in pure mathematics.

• A reflection in math is similar to a mirror image, where there’s a flip in the object over a line, which is also known as the line of reflection.

• After a reflection, the size and shape of the object remains the same. The object and its reflection are congruent to each other.

• The key characteristic of reflection is that it reverses the orientation of the object, for instance, a right-handed figure will be reflected as a left-handed figure.

• In terms of coordinate geometry, a reflection across the x-axis changes the sign of the y-coordinate, a reflection across the y-axis changes the sign of the x-coordinate, and a reflection on the origin changes the signs of both coordinates.

• Rotations, on the other hand, involve turning an object around a fixed point called the point of rotation, or the origin.

• Rotations are specified by the angle and direction (clockwise or anti-clockwise) of rotation.

• Similar to reflections, the size and shape of an object remains the same after rotations. The rotated object is also congruent to the original.

• In 2D coordinate plane, the coordinates of a point after rotation can be computed using rotation matrices.

• Rotation by 90° about the origin: (x, y) becomes (-y, x) Rotation by 180° about the origin: (x, y) becomes (-x, -y) Rotation by 270° about the origin: (x, y) becomes (y, -x)

• Understanding various properties of reflections and rotations is crucial for solving complex problems in mathematics, particularly in the field of geometry and trigonometry.

• It is important to practice plenty of problems based on reflections and rotations in order to get a better grasp of these concepts, their application and associated computation.