# EXPLORING ROTATIONS

## About "Exploring rotations"

Exploring rotations :

A rotation is a transformation that turns a figure around a given point called the center of rotation. The image has the same size and shape as the pre-image.

## Exploring rotations

The triangle XYZ has the following vertices.

X(0, 0), Y(2, 0) and Z(2, 4).

(i)  Trace triangle xyz and the x- and y-axes onto a piece of paper. (ii) Rotate your triangle 90° counterclockwise about the origin. The side of the triangle that lies along the x-axis should now lie along the y-axis.

(iii) Sketch the image of the rotation. Label the images of points X, Y, and Z as X', Y', and Z'. (iv)  Write the vertices of the image x'y'z'.

X'(0, 0), Y'(0, 2) and Z'(-4, 2)

(v) Rotate your triangle 90° clockwise about the origin. The side of the triangle that lies along the x-axis should now lie along the y-axis.

(vi) Sketch the image of the rotation. Label the images of points X, Y, and Z as X", Y", and Z". (vii)  Write the vertices of the image x"y"z".

X"(0, 0), Y"(0, -2) and Z"(4, -2)

(viii) Based on the rotations done above, write the general rule for 90° counterclockwise rotation about the origin.

X(0, 0) -------> X'(0, 0)

Y(2, 0) -------> Y'(0, 2)

Z(2, 4) -------> Z'(-4, 2)

From the above transformations of vertices, we have the following general rule for 90° counterclockwise rotation.

 Pre-image(x, y) Image (After rotation)(-y, x)

(ix) Based on the rotations done above, write the general rule for 90° clockwise rotation about the origin.

X(0, 0) -------> X"(0, 0)

Y(2, 0) -------> Y"(0, -2)

Z(2, 4) -------> Z"(4, -2)

From the above transformations of vertices, we have the following general rule for 90° clockwise rotation.

 Pre-image(x, y) Image (After rotation)(y, -x)

Similarly, we can define general rules for clockwise and counterclockwise rotations of 180° and 270°  about the origin as given below. ## Reflect

How are the size and the orientation of the triangle affected by the rotation ?

The size stays the same, but the orientation changes in that the triangle is turned or tilted left – what was “up” is now “left”. After having gone through the stuff given above, we hope that the students would have understood "Rotations".

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