TRANSFORMATIONS   

Transformation(A) Identifying a transformation

1.   A transformation is a process of moving a figure to a new position in a one-to-one                   correspondence or mapping between points of the original figure to the figure in the new       position in a plane.

2.   When a figure moved to a certain position, the original figure is called the object and the         figure in the new positon is called the image.

In the diagram above, figure B is moved to figure C or we can say  that figure B is mapped onto figure C.  Therefore, figure C is the image of figure B.

TRANSLATION

(A)   Identifying a Translation

Translation is a type of transformation that moves every point on a plane in the same direction and through the same distance.

Determine whether each of the following transformations is a translation.

(a)                                                                         (b)

                           

(c)

         

 (a)   Not a translation (b)   A translation (c)   Not a translation

(B)  Determining the Image of an Object under a Translation

In the diagram, determine the image of point T under a translation represented by line PQ.

Image of point T under a translation represented by line PQ is point C.

In the diagram, point M is the image of point J under a translation.  Determine the image of quadrilateral PQRS under the same translation.

Quadrilateral P' Q' R' S' is the image of quadrilateral PQRS under the same translation.

(C)   Describing a Translation

A translation can be described in two ways.

In the following diagrams, figure I is the image of figure II under a translation.  Describe the translation by stating the direction and distance.

(a)  

                                                                                                                

(b)

      

(a)   The translation is a movement through a distance of 7 units vertically upwards.(b)   The translation is a movement through a distance of MN in the direction from M to N.

In the diagram, point M is the image of point J under a translation.  Determine the image of quadrilateral PQRS under the same translation.

(D)  Properties of Translation

Under a translation, the shape, size and orientaiton of the object and the image remain the same.Example :

In the diagram, triangle F' J' L' is the image of triangle FJL under a translation.

(E)  Determining the Coordinates of the Image and the Object under a Translation

In the diagram, P' is the image of point P under a translation.  Determine the coordinates of images of points Q and R under the same translation.

From the diagram, the coordinates of images of points Q and R are Q' (8, 9) and R' (4, 8) respectively.

 

(F)  Solving Problem Involving Translation

The graph show three points, H, H' and L'.

 (a)

        (b)   To locate point L, move point L' 3 units to the left parallel to the x-axis and then 2 units           upwards parallel to the y-axis.  Coordinates of L (from the graph) is (1, 6)

  We can also use the following method to find the coordinates of L.

  

REFLECTION

(A)   Identifying a reflection

A reflection is a transformation where all the points in a plane are inverted or reflected to the other side in a line called axis of reflection.For example :

Triangle D is reflected in the line ST and D' is the image.

(B)   Determining the image of an object under a reflection in a given line.

By using a piece of tracing paper or by paper folding, we can find the image of an object under a reflection.

(C)   Determining the properties of reflection

In the diagram, P'Q'RS'T' is the image of PQRST under a reflection.  The properties of reflection :

(a)   The object and its image have the same shape and size.(b)   The orientation of the image is laterally inverted as compared to that of the object.(c)   The points on the axis of reflection do not change in position.(d)   The axis of reflection is a perpendicular bisector of the line joining the object and its               image.

(D)   Determining the image of an object and the axis of reflection

Construct the image of the triangle TUV below under a reflection in the straight line EF.

 (i)   Using a pair of compasses, with centre U, draw two arcs which cross EF.

 (ii)  Use each of the intersecting points in (i) as centre to mark arcs that intersect at U'.

Determine the axis of reflection for the object and its image.

(i)   Choose any points on the object, say N, and join it to its image, N'.

(ii)   Construct a perpendicular bisect to the line NN'.

(iii)  The perpendicular bisector LM is the axis of reflection.

(E)   Determining the coordinates of the image and object under a reflection

To determine the coordinates of the images, given the coodinates of the object.

 Determine the coodinates of the image of point E(4, 2) under a reflection in the (a)   x-axis (b)   y-axis

 (a)   Point E is 2 units above the x-axis.  Its image is of the same distance below the x-axis.         Therefore, under a reflection in the x-axis, the image of E(4, 2) is E'(4, -2).

 (b)   Point E is 4 units to the right of the y-axis.  Its image is of the same distance to the left of       the y-axis.  Therefore, under a reflection in the y-axis, the image of E(4, 2) is E"(-4, 2).

To determine the coordinates of the object , given the coodinates of the image

To describe a reflection, state the axis of the reflection.

Draw a perpendicular line from C'(2, 1) to the line ST.  The distance from C to the line ST is the same as the distance from C' to line ST.  From the diagram, the coordinates of C are (-2, 5).

(F)   Describing a reflection given the object and image

To describe a reflection, state the axis of the reflection.

In the diagram, triangle UVW is the image of triangle UVT under a reflection.  Describe the reflection.

Reflection in the line UV.

(G)   Solving problems involving reflections

A student stood in front of a mirror.  On the wall behind his back, there was a sign 'KEEP QUIET'. What is the image of the sign seen on the mirror by the student?

ROTATION

(A) Identifying a Rotation

A rotation is a transformation that turns all points on a plane about a fixed point through an angle in either clockwise or anticlockwise direction.  The fixed point is known as the centre of rotation and the angle is known as the angle of rotation.

(a)                                                                          (b)  

                               

(c)

(a)   Not a rotation

(b)   Not a rotation

(c)   A rotation

(B) Determining the Image of an Object under a Rotation

Determine the image of line AB under a rotation about point O through 90° in the following direction.(a)   Clockwise                                                   (b)   Anticlockwise

(a)  

(b)

(C) Properties of Rotation

(a)   Each point is turned through the same angle.(b)   The distances of a point and its image from the centre of rotation area the same.

(c)   The centre of rotation is the only point that does not change its position under a rotation.(d)   Under a rotation, the shapes, sizes and orientations of the object and its image are the same.

(D) Determining the Image of an Object Given the Centre, Angle and Direction of Rotation

In the diagram, triangle ABC is rotated about point O through 180°.(a)   Determine the image of triangle ABC under the rotation.(b)   Is there any difference between the shapes, sizes and orientations of triangle ABC and           its image under the rotation?

(a)

(b)   The shapes, sizes and orientations of triangle ABC and its image under the rotation are         the same.

(E) Determining the Centre, Angle and Direction of Rotation

In the diagram, triangle P'Q'R' is the image of triangle PQR under a rotation.  Determine the centre, angle and the direction of the rotation.

Determine the perpendicular bisectors of the line joining point P and its image P', point R and its image R', or point Q and its image Q'.  The centre of the rotation, O, is the point of intersection of any two perpendicular bisectors.

The rotation has centre O with the angle of rotation 90° the anticlockwise direction.

(F) Determining the Coordinates of the Image

In the diagram, triangle PQR is rotated about the origin O through 90° anticlockwise. Determine the coordinates of the images of points P, Q and R.

(G) Determining the Coordinates of the Object

In the above diagram, triangle F'H'J' is the image of triangle FHJ under a rotation.(a)   Describe the rotation by sating its centre, angle and direction of rotation.(b)   Given that point N is the image of point M under the same rotation, determine the                   coordinates of M.

(a)

(b)   The coordinates of point M is (8, 6).

(H) Solving Problems Involving Rotation

The coordinates of the vertices of the image are S'(-4, 1), T'(-4, 3) and U'(-2, 0).

ISOMETRY

(A) Identifying a isometry1.   An isometry is a transformation that preserves the shape and size of the object.2.   'ISO' means same and 'metry' means measurement.  Therefore, under an isomety, the           image is identical to the object in its shape and size.3.   Examples of isometric transformation are translation, reflection and rotation as the shape     and size of the object is unchanged.4.   A combination of two or more isometric transformations is also an isometry.For example,

(B) Determining whether a given transformation is an isometry

Figures I, II and III are the images of the figure M under a reflection, a rotation and a translation respectively.  Determine whether each transformation is an isometry.

(i)   Trace the figure M onto a tracing paper.

(ii)  Match the traced figure to each image.

The traced figure fits each image exactly.  Therefore, reflection, rotation and translation are isometric transformations.

(C)   Constructiong patterns using isometry

The idea of isometry can be used to form patterns and designs.For example,

CONGRUENCE(A) Identifying Congruent Figures

Congruent figures are figures that have the same size and shape regardless of their orientations.

(B) Identifying Congruency between Two Figures As a Propertiy of an Isometry

In the diagram, trapezium PRTW is the image of trapezium FGJN under an isometric transformation.(a)   Determine the isometric transformation.(b)   What can you say about the congruency of the two trapeziums?

(C)   Solving Problems Involving Congruence

PROPERTIES OF QUADRILATERALS

(A) Determining the properties of a quadrilateral using reflection and rotation

1.   Reflection and rotation can be used to determine the properties of a quadrilateral in                 relation to its sides, angles and diagonals.

2.   The properties of quadrilaterals are as follows :(a)   Square

        (i)   The length of each side is equal.        (ii)  The opposide sides are parallel.        (iii) All angles are 90°.        (iv) Both diagonals are equal and bisect each other at 90°.

(b)   Rectangle

         (i)   The opposite sides are equal and parallel.         (ii)  All angles are 90°.         (iii) Both diagonals are equal and bisect each other.

(c)   Rhombus

          (i)   The length of each side is equal.          (ii)  The opposite sides are parallel.         (iii)  The opposite angles are equal.         (iv)  Both diagonals bisect each other at 90°.

(d)   Parallelogram

         (i)   The opposite sides are equal and parallel.         (ii)  The opposite angles are equal.         (iii) Both diagonals bisect each other.

(e)   Kite

   (i)  Two pairs of equal adjacent sidesl.  (ii)  Only a pair of opposite angles are equall. (iii)  The longer diagonal is the perpendicular bisector of the shorter diagonal. (iv)  The longer diagonal bisect a pair of opposite angles.

In the diagram, rectangle A'B'C'D' is the image of rectangle ABCD under an isometry.(a)   Describe the isometry.(b)   If point R is the image of point P under the same isometry, determine the coordinates of         point P.

The isometry is a rotation about point T.

(a)   The isometry is a rotation about point T(2, 1) through 90° clockwise.(b)   The coordinates of point P are (6, 8).

In the diagram on the right, PQ is a straight line.  Which of the points A, B, C or D is the image of the point T under a reflection in the line PQ?

A in not the image of T under a reflection in the line PQ because the axis of reflection PQ is not the perpendicular bisector of the line AT.

The point C is the image of the point T under a reflection in the line PQ.

   

     

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