Image The figure that results from a transformation. Pre-image The - PowerPoint PPT Presentation

D AY 7 – F IGURE - PRESERVING SERIES OF ROTATIONS AND REFLECTIONS

I NTRODUCTION In art and design, we see objects rotated to different positions to enhance visual appearance. We see reflections of objects in water, in a mirror and in glass to form images. In this lesson, we are going to discuss figure-preserving series of rotations and reflections, where the image is mapped onto the pre-image. A plane figure is mapped onto itself when a series of transformation results in the original object.

V OCABULARY Transformation A change in either size, position, shape or both of a plane figure. Conventionally, we name images using the by adding a prime ( ′ ) to same letters used to name the pre-image. Rotation A transformation which turns a plane figure about a fixed point called the center of rotation . Reflection A transformation where a plane figure is reflected about a line, called the line of reflection or the mirror line to form an image.

Image The figure that results from a transformation. Pre-image The original figure that has not been transformed, also called the object. Mid-point A point exactly at center of a line segment that divides the segment into two equal segments.

ROTATION In rotation, the center of rotation is fixed while every point of the pre-image is turned about a fixed point called the center of rotation . A plane figure can be rotated about its center until it fits onto itself. This can occur two or more times in one turn depending on the shape of the figure. One turn is equivalent to 360°. A figure can be rotated either clockwise or anti- clockwise. Clockwise rotation is a rotation in negative angle direction while anti-clockwise rotation is in positive angle direction.

A rotation that maps a rectangle onto itself In rectangle ABCD below, we can come up with a series of rotations that map it onto itself. First, we locate its center O by drawing the diagonals AC and BD which intersect at the center. D A O B C

We rotate the rectangle clockwise through an angle of 90° about the center. The series of rotations is shown below. A B D A 90° B C C D

We rotate the rectangle again from the new position through an angle 90° . The object has been rotated through a total of 180° from its original position. A B C B 90° A D C D The image formed is mapped onto the object.

If we continue rotating it for another 90° ( to a total 270 ° ), we have: D C C B 90° A D B A

If we rotate the previous image through 90° ( to a total of 360° ), we have: D C D A 90° B C A B This maps the image onto the pre-image again.

We now discover that after a rotation of 180° and 360° about its center, the rectangle is mapped exactly onto itself. It fits into itself twice in a rotation of 360° .

A rotation that maps a parallelogram onto itself Given the parallelogram PQRS we locate the center by drawing diagonals which intersect at its center and rotate it about the center through 90° intervals until it is mapped onto itself. The series of clockwise rotations is shown below. Q S P P 90° R Q R S

T he next rotation of 90° Q Q R P P S R S We observe that the image formed is mapped onto the pre-image after a rotation of from the original position 180°

After a rotation of 270° from the original position, we have: S Q R R P S P Q

After a rotation of 360° from the original position, we have: S S R P R Q P Q The image is mapped onto the pre-image again.

Just like the rotation of a rectangle about its center, the parallelogram is mapped onto itself twice in a turn, after a rotation of 180° and 360° respectively.

A rotation that maps a trapezoid onto itself The trapezoid undergoes a series of four 90° − rotation about the center to get the image mapped onto the pre-image. The center is located as shown. After a rotation of 90° about the center, we have: L K N K M L N M

After a rotation of 180° about the center, we have: L K L M K N N M

After a rotation of 270° about the center, we have: M L M N K N K L

After a rotation of 360° , the image is mapped onto the pre-image for the first time. M N K N M L K L

A rotation that maps a regular polygon onto itself A regular polygon is a plane figure having all angles and all sides equal. We can determine the angle of rotation about the center of any regular polygon that maps the image onto the pre-image by the formula: 360° 𝑜 where n is the number sides of the polygon.

A rotation that maps a regular hexagon onto itself A hexagon has six sides and therefore the angle of rotation will be: 360° = 60° 6 Multiple of 60° , that is 120° or 180° about its center will map the regular hexagon onto itself. So the hexagon will be mapped onto itself in a series of rotations at intervals of 60° either clockwise or anticlockwise.

After a every rotation of 60º, the image is mapped onto the pre-image. 60° O O

Rotation about an external center to a plane figure A rotation of 360º about a given center outside a plane figure will always map the image onto the pre-image. The rotation can be either clockwise or anticlockwise. For example if a rectangle is rotated through an angle of 360° in either direction, the image is mapped onto the pre-image as shown below. The series of rotations after 360° intervals such as 720°, 1080° will also map the image onto the pre- image.

Every point on the rectangle is rotated at an angle of 360° about the center. The rectangle is mapped onto itself as shown. 𝐸 𝐸 𝐵 𝐵 𝐷 𝐶 𝐷 𝐶 360°

R EFLECTION The line of reflection helps us to reflect a pre-image onto itself. If we reflect a pre-image of any plane figure onto the image then treat the image as the pre-image and reflect it again using the same line of reflection, the pre-image is mapped back onto the object.

A rectangle is reflected onto its image, then the image treated as the pre-image then reflected along the same line of reflection. The image gets mapped onto the object. Line of reflection Pre- Image image Pre- image Image

This means that even reflections such as the second, fourth or sixth will map the image back to the pre-image along the same line of reflection. We can use the same procedure for other quadrilaterals like a parallelogram, trapezoid and other regular polygons.

Reflections along specific mirror lines on some symmetrical plane figures maps the image onto the pre-image. For example in a rectangle, reflections along the dotted mirror lines will result in the image being mapped onto the pre-image.

A rectangle is mapped onto itself along the mirror lines 𝑚 1 and 𝑚 2 passing through the midpoints of its sides. 𝑚 1 𝑚 2

Rectangle MNOP is mapped onto itself when reflected along mirror line 𝑚 2 as shown below. The vertices of the image have the prime symbol. 𝑂M 𝑃P 𝑚 2 𝑄O 𝑁 ́N

The isosceles trapezoid and rhombus are mapped onto the themselves when reflected through the dotted mirror lines shown below.

For regular polygons, the lines of reflection that map the image back to the object are equal to the number of sides of the polygon. For instance, in a regular pentagon, there are five lines of reflection that will map the polygon onto itself.

HOMEWORK Find the least angle of rotation that a regular pentagon can be rotated about its center to get mapped onto itself.

A NSWERS TO HOMEWORK 72°

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