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KS3 Mathematics

KS3 Mathematics. S4 Coordinates and transformations 1. Reflection. An object can be reflected in a mirror line or axis of reflection to produce an image of the object. For example,.

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KS3 Mathematics

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  1. KS3 Mathematics S4 Coordinates and transformations 1

  2. Reflection An object can be reflected in a mirror line or axis of reflection to produce an image of the object. For example, Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.

  3. Reflecting shapes A B C D If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’. A’ B’ object image C’ D’ mirror line or axis of reflection The image is congruent to the original shape.

  4. Reflecting shapes A A’ B B’ object image C C’ D D’ mirror line or axis of reflection If we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line.

  5. Reflection on a coordinate grid A’(–2, 6) B’(–7, 3) C’(–4, –1) y The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). A(2, 6) 7 6 5 B(7, 3) 4 3 2 Reflect the triangle in the y-axis and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 C(4, –1) –3 –4 What do you notice about each point and its image? –5 –6 –7

  6. Reflection on a coordinate grid B’(–1, 7) C’(–6, 2) y x = y The vertices of a triangle lie on the points A(4, 4), B(7, –1) and C(2, –6). 7 6 A’(4, 4) 5 4 A(4, 4) 3 2 Reflect the triangle in the line y = x and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x B(7, –1) –1 –2 –3 –4 What do you notice about each point and its image? –5 –6 C(2, –6) –7

  7. Planes of symmetry Is a cube symmetrical? We can divide the cube into two symmetrical parts here. This shaded area is called a plane of symmetry. How many planes of symmetry does a cube have?

  8. Planes of symmetry Is a cube symmetrical? We can divide the cube into two symmetrical parts here. This shaded area is called a plane of symmetry. How many planes of symmetry does a cube have?

  9. Rotational symmetry An object has rotational symmetry if it fits exactly onto itself when it is turned about a point at its centre. The order of rotational symmetry is the number of times the object fits onto itself during a 360° turn. If the order of rotational symmetry is one, then the object has to be rotated through 360° before it fits onto itself again. Only objects that have rotational symmetry of two or more are said to have rotational symmetry. We can find the order of rotational symmetry using tracing paper.

  10. Rotational symmetry What is the order of rotational symmetry for the following designs? Rotational symmetry order 4 Rotational symmetry order 3 Rotational symmetry order 5

  11. Describing a rotation The angle of rotation. The direction of rotation. The centre of rotation. A rotation occurs when an object is turned around a fixed point. To describe a rotation we need to know three things: For example, ½ turn = 180° ¼ turn = 90° ¾ turn = 270° For example, clockwise or anticlockwise. This is the fixed point about which an object moves.

  12. Rotating shapes If we rotate triangle ABC 90° clockwise about point O the following image is produced: B object 90° A A’ image B’ C C’ O A is mapped onto A’, B is mapped onto B’ and C is mapped onto C’. The image triangle A’B’C’ is congruent to triangle ABC.

  13. Rotating shapes The centre of rotation can also be inside the shape. For example, 90° O Rotating this shape 90° anticlockwise about point O produces the following image.

  14. Translation A’ image A A A A A A A A C’ B’ object object object object object object object object C C C C C C C C B B B B B B B B When an object is moved in a straight line in a given direction we say that it has been translated. For example, we can translate triangle ABC 5 squares to the right and 2 squares up. Every point in the shape moves the same distance in the same direction.

  15. Translations When a shape is translated the image is congruent to the original. The orientations of the original shape and its image are the same. An inverse translation maps the image that has been translated back onto the original object. What is the inverse of a translation 7 units to the left and 3 units down? The inverse is an equal move in the opposite direction. That is, 7 units right and 3 units up.

  16. Describing translations 3 For example, the vector describes a translation 3 right and 4 down. –4 When we describe a translation we always give the movement left or right first followed by the movement up or down. We can also describe translations using vectors. As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left. One more way of describing a translation is to give the direction as an angle and the distance as a length.

  17. Translations on a coordinate grid y The coordinates of the vertex A of this shape are (–4, –2). 7 6 5 4 3 When the shape is translated the coordinates of the vertex A’ are (3, 2). 2 A’(3, 2) 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 What translation will map the shape onto its image? A(–4, –2) –3 –4 –5 –6 –7 7 right 4 up

  18. Translations on a coordinate grid y The coordinates of the vertex A of this shape are (3, –4). 7 6 5 4 3 When the shape is translated the coordinates of the vertex A’ are(–3, 3). A’(–3, 3) 2 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 What translation will map the shape onto its image? –3 –4 A(3, –4) –5 –6 –7 6 left 7 up

  19. Congruence and similarity Is the image of an object that has been enlarged congruent to the object? Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal. In an enlarged shape the corresponding angles are the same but the lengths are different. The image of an object that has been enlarged is not congruent to the object, but it is similar. In maths, two shapes are called similar if their corresponding angles are equal and their corresponding sides are different but in the same ratio.

  20. Enlargement A’ A Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. We say that shape A has been enlarged by scale factor 2.

  21. Find the scale factor What is the scale factor for the following enlargements? D D’ Scale factor = 0.5

  22. Enlargement on a coordinate grid A’(4, 8) C’(8, 6) B’(6, 2) y The vertices of a triangle lie at the points A(2, 4), B(3, 1) and C(4, 3). 10 9 8 7 The triangle is enlarged by a scale factor of 2 with a centre of enlargement at the origin (0, 0). 6 5 A(2, 4) 4 C(4, 3) 3 2 1 B(3, 1) What do you notice about each point and its image? 0 1 2 3 4 5 6 7 8 9 10 x

  23. Enlargement on a coordinate grid y 10 A(6, 9) C’(9, 9) 9 8 7 6 5 4 A(2, 3) 3 C(3, 3) B’(6, 3) 2 1 B(2, 1) 0 1 2 3 4 5 6 7 8 9 10 x The vertices of a triangle lie at the points A(2, 3), B(2, 1) and C(3, 3). The triangle is enlarged by a scale factor of 3 with a centre of enlargement at the origin (0, 0). What do you notice about each point and its image?

  24. Negative scale factors This example shows the shape ABCD enlarged by a scale factor of –2 about the centre of enlargement O. D’ A B O C C’ D B’ A’ When the scale factor is negative the enlargement is on the opposite side of the centre of enlargement.

  25. Enlargement on a coordinate grid

  26. Finding the centre of enlargement Find the centre the enlargement of A onto A’. This is the centre of enlargement A’ A • Draw lines from any two vertices to their images. • Extend the lines until they meet at a point.

  27. Finding the centre of enlargement Find the centre the enlargement of A onto A’. A A’ • Draw lines from any two vertices to their images. • When the enlargement is negative the centre of enlargement is at the point where the lines intersect.

  28. Describing enlargements

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