© boardworks ltd 2004 1 of 58 ks3 mathematics s4 coordinates and transformations 1
TRANSCRIPT
© Boardworks Ltd 2004 1 of 58
KS3 Mathematics
S4 Coordinates and transformations 1
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Contents
S4 Coordinates and transformations 1
A
A
A
A
AS4.1 Coordinates
S4.5 Rotation symmetry
S4.4 Rotation
S4.2 Reflection
S4.3 Reflection symmetry
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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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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 1 2 3 4–4 –3 –2 –1
1
2
3
4
–4
–3
–2
–1
x-axis
y-axis
origin
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first quadrant
second quadrant
fourth quadrant
third quadrant
0 1 2 3 4–4 –3 –2 –1
1
2
3
4
–4
–3
–2
–1
Quadrants
The coordinate axes divide the grid into four quadrants.
y
x
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Which quadrant?
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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.Along the corridor and up (or down) the stairs.
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Plotting points
Plot the point (–3, 5).
0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1
(–3, 5)
x
y
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Plotting points
0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1
(–4, –2)
Plot the point (–4, –2).
x
y
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0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
y
Plotting points
(6, –7)
Plot the point (6, –7).
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0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
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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Making quadrilaterals
Where could we add a fourth point to make a square?
0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
y
(6, 2)
(2, 6)
(2, –2)
(–2, 2)
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0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
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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0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
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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0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
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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0 1 2 3 4 5 6 7–1–2–3–4–5–6–7
1
2
3
4
5
6
7
–2
–4
–6
–3
–5
–7
–1x
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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Don’t connect three!
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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(xm, 5).
xm is the point half-way between –2 and 6.
A(–2, 5) B(6, 5)?M(xm, 5).
?8
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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, xm = –2 + ½ × 8
A(–2, 5) B(6, 5)?M(xm, 5).
?8
= –2 + 4
= 2
or xm = ½(–2 + 6)
= ½ × 4
= 2
The coordinates of the mid-point of AB are (2, 5).
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The x-coordinate of the point A is 2The 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.
0 1 2 3 4 5 6 7 8 9 10
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 + 82
= 5x
y
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and the y-coordinate of the point B is 5.
0 1 2 3 4 5 6 7 8 9 10
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 + 52
= 3
The mid-point of AB is (5, 3).
M(5, 3)
x
y
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Finding the mid-point of a line
If the coordinates of A are (x1, y1) and the coordinates of B are (x2, y2) 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.
x1 + x2
2is the mean of the x-coordinates.
x1 + x2
2,y1 + y2
2
y1 + y2
2is the mean of the y-coordinates.
x1 + x2
2,y1 + y2
2 B(x2, y2)
A(x1, y1)x
y