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© Boardworks Ltd 2004 1 of 58 KS3 Mathematics S4 Coordinates and transformations 1.

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Presentation on theme: "© Boardworks Ltd 2004 1 of 58 KS3 Mathematics S4 Coordinates and transformations 1."— Presentation transcript:

1 © Boardworks Ltd 2004 1 of 58 KS3 Mathematics S4 Coordinates and transformations 1

2 © Boardworks Ltd 2004 2 of 58 Contents S4 Coordinates and transformations 1 A A A A A S4.1 Coordinates S4.5 Rotation symmetry S4.4 Rotation S4.2 Reflection S4.3 Reflection symmetry

3 © Boardworks Ltd 2004 3 of 58 Coordinates We can describe the position of any point on a 2-dimensional plane using coordinates. The coordinate of a point tells us where the point is relative to a starting point or origin. For example, when we write a coordinate the first number is called the x -coordinate and the second number is called the y -coordinate. (3, 5) x -coordinate (3, 5) y -coordinate (3, 5) the first number is called the x -coordinate and the second number is called the y -coordinate.

4 © Boardworks Ltd 2004 4 of 58 Using a coordinate grid Coordinates are plotted on a grid of squares. The x -axis and the y -axis intersect at the origin. The coordinates of the origin are (0, 0). The lines of the grid are numbered using positive and negative integers as follows. 0 1234–4–3–2–1 1 2 3 4 –4 –3 –2 –1 x -axis y -axis origin

5 © Boardworks Ltd 2004 5 of 58 first quadrant second quadrant fourth quadrant third quadrant 0 1234–4–3–2–1 1 2 3 4 –4 –3 –2 –1 Quadrants The coordinate axes divide the grid into four quadrants. y x

6 © Boardworks Ltd 2004 6 of 58 Which quadrant?

7 © Boardworks Ltd 2004 7 of 58 Coordinates The first number in the coordinate pair tells you how many units along from the origin the point is in the x -direction. A positive number means the point is right of the origin and a negative number means it is left. The second number in the coordinate pair tells you how many units above or below the origin the point is in the y - direction. A positive number means the point is above the origin and a negative number means it is below. Remember: Along the corridor and up (or down) the stairs.

8 © Boardworks Ltd 2004 8 of 58 Plotting points Plot the point (–3, 5). 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 (–3, 5) x y

9 © Boardworks Ltd 2004 9 of 58 Plotting points 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 (–4, –2) Plot the point (–4, –2). x y

10 © Boardworks Ltd 2004 10 of 58 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y Plotting points (6, –7) Plot the point (6, –7).

11 © Boardworks Ltd 2004 11 of 58 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y Making quadrilaterals Where could we add a fourth point to make a parallelogram? (3, –3) (–5, –1) (–5, 4) (3, 2)

12 © Boardworks Ltd 2004 12 of 58 Making quadrilaterals Where could we add a fourth point to make a square? 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y (6, 2) (2, 6) (2, –2) (–2, 2)

13 © Boardworks Ltd 2004 13 of 58 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y Making quadrilaterals Where could we add a fourth point to make a rhombus? (–7, 2) (3, 2) (–2, 0) (–2, 4)

14 © Boardworks Ltd 2004 14 of 58 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y Making quadrilaterals Where could we add a fourth point to make a kite? (5, –1) (2, 2) (–7, –1) (2, –4)

15 © Boardworks Ltd 2004 15 of 58 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y Making quadrilaterals Where could we add a fourth point to make an arrowhead? (3, –2) (3, 3) (6, 6) (0, 6)

16 © Boardworks Ltd 2004 16 of 58 01234567–1–2–3–4–5–6–7 1 2 3 4 5 6 7 –2 –4 –6 –3 –5 –7 –1 x y Making quadrilaterals Where could we add a fourth point to make a rectangle? (–3, –3) (2, 7) (5, 1) (–6, 3)

17 © Boardworks Ltd 2004 17 of 58 Don’t connect three!

18 © Boardworks Ltd 2004 18 of 58 Finding the mid-point of a horizontal line Two points A and B have the same y -coordinate. A is the point (–2, 5) and B is the point (6, 5). What is the coordinate of the mid-point of the line segment AB? Let’s call the mid-point M( x m, 5). x m is the point half-way between –2 and 6. A(–2, 5)B(6, 5) ?M( x m, 5). ?8

19 © Boardworks Ltd 2004 19 of 58 Finding the mid-point of a horizontal line Two points A and B have the same y -coordinate. A is the point (–2, 5) and B is the point (6, 5). Either, x m = –2 + ½ × 8 A(–2, 5)B(6, 5)?M( x m, 5). ?8 = –2 + 4 = 2 or x m = ½(–2 + 6) = ½ × 4 = 2 The coordinates of the mid-point of AB are (2, 5).

20 © Boardworks Ltd 2004 20 of 58 The x -coordinate of the point A is 2 The x -coordinate of the point A is 2 and the x -coordinate of the point B is 8. Finding the mid-point of a line If A is the point (2, 1) and B is the point (8, 5), what is the mid-point of the line AB? Start by plotting points A and B on a coordinate grid. 012345678910 1 2 3 4 5 6 7 B(8, 5) A(2, 1) The x -coordinate of the mid- point is half-way between 2 and 8. 2 + 8 2 = 5 x y

21 © Boardworks Ltd 2004 21 of 58 and the y -coordinate of the point B is 5. 012345678910 1 2 3 4 5 6 7 B(8, 5) A(2, 1) Finding the mid-point of a line If A is the point (2, 1) and B is the point (8, 5), what is the mid-point of the line AB? Start by plotting points A and B on a coordinate grid. The y -coordinate of the point A is 1 The y -coordinate of the mid- point is half-way between 1 and 5. 1 + 5 2 = 3 The mid-point of AB is (5, 3). M(5, 3) x y

22 © Boardworks Ltd 2004 22 of 58 Finding the mid-point of a line If the coordinates of A are ( x 1, y 1 ) and the coordinates of B are ( x 2, y 2 ) then the coordinates of the mid-point of the line segment joining these points are given by: We can generalize this result to find the mid-point of any line. x 1 + x 2 2 is the mean of the x -coordinates. x 1 + x 2 2, y 1 + y 2 2 2 is the mean of the y -coordinates. x 1 + x 2 2, y 1 + y 2 2 B( x 2, y 2 ) A( x 1, y 1 ) x y


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