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

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© Boardworks Ltd 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

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© Boardworks Ltd 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.

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© Boardworks Ltd 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 –4–3–2– –4 –3 –2 –1 x -axis y -axis origin

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

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© Boardworks Ltd of 58 Which quadrant?

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© Boardworks Ltd 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.

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

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

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

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© Boardworks Ltd of –1–2–3–4–5–6– –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)

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

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© Boardworks Ltd of –1–2–3–4–5–6– –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)

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© Boardworks Ltd of –1–2–3–4–5–6– –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)

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© Boardworks Ltd of –1–2–3–4–5–6– –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)

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© Boardworks Ltd of –1–2–3–4–5–6– –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)

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© Boardworks Ltd of 58 Don’t connect three!

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© Boardworks Ltd 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

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© Boardworks Ltd 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).

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© Boardworks Ltd 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 B(8, 5) A(2, 1) The x -coordinate of the mid- point is half-way between 2 and = 5 x y

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© Boardworks Ltd of 58 and the y -coordinate of the point B is 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 = 3 The mid-point of AB is (5, 3). M(5, 3) x y

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© Boardworks Ltd 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 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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