Day 9 – Pre-image and Image under transformation

Introduction Having discussed what transformations are, it is always necessary to practice how to carry out them. We will apply the concepts and the properties of each transformation to do carry out the transformations on a graph paper. We will also try to identify the sequence of transformations used to arrive at the image.

Vocabulary Transformations This refers to a change in shape and position of an image from one form or location to another. Perpendicular lines They are lines that intersect each other at 90°. Parallel lines They are lines that are separated by the same distance along their length, hence, they do not meet.

Translation Given a figure, we can determine its image under translation is the direction of translation is shown. We will demonstrate this by an example. Example 1. Given a triangle ABC, draw its image if it is translated 7 units to the right and 2 units downwards. Label the image as 𝐴 ′ 𝐵 ′ 𝐶 ′ .

Solution From each point, A,B and C, we draw lines that passes through a point that is 7 units to the left and units each with respect to each point. A B C

Upon drawing, we get the image 𝐴 ′ 𝐵 ′ 𝐶 ′ . B C 𝐀′ 𝐁′ 𝐂′ C A B

Rotation To draw an image of an object about rotation, we first establish the center of rotation then connect all vertices to it. The then draw arcs whose radius is equal to the distance from the center of rotation to the vertex. The central angle of the arcs must exceeds slightly to the angle or rotation given. We then measure the angle given from the line from the center of rotation to the vertex then locate the image along the arc. We do that for other vertices too then connect the new points.

Example 2 Given the figure below, draw its image under rotation of 90° about the point given near the image.

Solution We connect the three vertices to the point using a line Solution We connect the three vertices to the point using a line. We draw an arc (with center at the point) of central angle more than 90° from the vertices to the left. We then mark the points on the marks at which the angle is 90°. We then connect the points to have the following diagram,

The resultant image is shown below

Reflection This refers to a flip of an object Reflection This refers to a flip of an object. Given an object, we can draw its image. From each vertex, we drop a perpendicular to the mirror line. Mark the image point to be at equal distance from the mirror line as the object point. We then connect the points to get the image. Let us have an example

Example 3 Given the following figure, draw its image under reference reflection about the line given.

Solution From each vertex, drop a perpendicular to the mirror line Solution From each vertex, drop a perpendicular to the mirror line. Mark the image point to be at equal distance from the mirror line as the object point. Connect the points to get the image

The image is as shown below

Sequence of transformations So far, we are familiar with a number of transformations, hence we should be able to tell a sequence of transformations between any two figures. We will only concentrate on translation, reflection and rotation. Example 4 Identify the transformations that map the two images (i) (ii)

Solution The long tip is facing in opposite direction of each other hence, that is a sign of reflection. Since the figures are not symmetrical about any line, one must have been translated after reflection. Thus, the transformations are reflection then translation or translation then reflection. Example 5 Identify the transformations that map the two images (i) (ii)

Solution The caved in part is facing upwards and not in opposite direction of each other hence, that is a sign of rotation. When we connect corresponding tips with a line, at least 3 pairs, the perpendicular bisector of these lines does not intersect at one point showing that the image was translated after rotation. Thus, the transformations are rotation then translation or translation then rotation.

homework Draw the image of the following figure under reflection about the given line.

Answers to homework The image is as shown below

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