2.1 Coordinate geometry:
Theory notes
2.1 Coordinate Geometry (AS91256)
AS91256 Internal, 2020
4 weeks/2 credits
Department of Mathematics
Weekly planner
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Weekly planner
1.6 Apply Geometric Reasoning (AS91031)
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Builds on what you learned in year 9 to 11
2.1 Coordinate Geometry (AS91256)
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Algebra/Number
Geometry
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Coordinate geometry connects the material in 1.6 Geometry with 1.3 Tables, Equations and Graphs.
Basic vocabulary
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Coordinate geometry is geometry in the cartesian plane, a two dimensional number line.
A point is defined by its (x,y) coordinates.
A line is defined by the equation y=mx+c.
A shape is defined by the gradients, distances and angles of its lines.
A gradient is the “steepness” of a line: how much the y-coordinates of the points on a line change with a unit change in their x-coordinates.
The 2.1 exams cover skills on three different levels
2.5 Networks (AS91260)
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(Week 1: midpoints, gradients and distances)
(Week 2: Right angled triangles, isosceles triangles, trapeziums, parallelograms)
(Week 3: Rectangles, right angled trapeziums, right angled isosceles triangles.)
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Topics
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3rd week: Shapes
4th week: Revision
2nd week: Shapes
1st week: Lines
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Expectations
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What to bring
What to do
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Week 1: Midpoints, gradients and distances
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Midpoints
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What is the midpoint of these two points?
Midpoints
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What is the midpoint of these two points?
The midpoint between (0,4) and (9,0) is exactly halfway between the x-values 0 and 9 and the y-values between 4 and 0.
This means that the midpoint is the mean of each:
Distance between points
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What is the distance between these points?
Distance between points
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What is the distance between these points?
The distance between two points on the cartesian plane is determined by Pythagoras' theorem.
Distance between points
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What is the distance between these points?
The distance between two points on the cartesian plane is determined by Pythagoras' theorem.
Pythagoras’ theorem: Big picture
Distance between points
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What is the distance between these points?
If you know the position of any two points, Pythagoras' theorem gives you the distance between them.
Pythagoras’ theorem: Big picture
Distance between points
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What is the distance between these points?
If you know the position of any two points, Pythagoras' theorem gives you the distance between them.
Let’s express this as an equation.
Pythagoras’ theorem: Big picture
Distance between points
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The distance equation
Using this equation, the distance between A(0,4) and B(9,0) can be calculated as
A
B
Pythagoras’ theorem: Big picture
Distance between points
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So, the reason people care so much about Pythagoras’ theorem is not because they love triangles.
It is because it lets you calculate the distance between two points.
Calculating the distance is both easier and more accurate than the alternative: to directly try to somehow measure it.
Notation
2.1 Coordinate Geometry (AS91256)
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To make things easier for us, we often give our points name such as A and B followed by their (x,y) coordinates.
For example: If A has coordinates (3,2) and B has coordinates (2,4) we would simply write A(3,2) and B(2,4).
For the distance between the two points A and B points, we use the notation .
For example:
Gradient of a line
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What is the gradient of this line?
Gradient of a line
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What is the gradient of the line?
A gradient is the change in y for a unit increase in x.
Parallel lines
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Two lines are parallel if they have the same gradient.
Orthogonal/perpendicular line/meet at 90º
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Two lines are perpendicular if their gradients are each others negative reciprocal.
Week 2-3: Identifying shapes with reasons and evidence
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All answers in the exam need the following 4 parts
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Calculations
Name property
Reasoning
Gradients
Distances
Midpoints (as asked for)
What is the shape?
Triangles: Isosceles, right angled, equilateral.
Quadrilaterals: Rectangle, square, trapezium, kite, parallelogram, rhombus.
Pentagons
Right angled, regular
How do you know that the properties are there?
Show your evidence explicitly in connection to your conclusion and reasons.
Evidence
How is the shape defined?
Triangles: Isosceles, right angled.
Quadrilaterals: Rectangle, parallelogram, square, rhombus, kite.
Pentagons: Right angled, regular.
Definitions: triangles 1
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Shape | Definition | Suggested evidence | A point: Shapes with one defining property. M point: Shapes with two defining properties. |
Right angled triangle | 3 sides. 2 adjacent sides are perpendicular. | gAB×gBC=-1 | Achieved point |
Isosc. triangle | 3 sides. 2 same length. | |AB|=|BC|=value | Achieved point |
Right angled isosceles triangle | 3 sides. 2 adjacent sides perpendicular. 2 sides same length. | |AB|=|BC|=value gAB×gBC=-1 | Achieved point |
Equilateral triangle | All 3 sides have same length | |AB|=|BC|=|CA|| | Achieved point. |
Definitions: triangles 2
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Shape | Definition | Suggested evidence (without the numbers) | A point: Shapes with one defining property. M point: Shapes with two defining properties. |
Right angled triangle | 3 sides. 2 adjacent sides are perpendicular. | gAB×gBC=-1 | Achieved point |
Isosceles triangle | 3 sides. 2 sides same length. | |AB|=|BC|=value | Achieved point |
Right angled isosceles triangle | 3 sides. 2 adjacent sides are perpendicular. 2 sides have same length. | |AB|=|BC|=value gAB×gBC=value×value=-1 | Merit point |
Equilateral triangle | 3 sides of same length | |AB|=|BC|=|CA||=value | Achieved point. |
Definitions: Quadrilaterals 1
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Shape | Definition | Suggested evidence (without the numbers) | A point: Shapes with one defining property. M point: Shapes with two defining properties.. |
Parallelogram (Need to check if square or rectangle to get E) | 4 sides. All opposite sides are parallel. All opposite sides are equal length. | gAB=gCD=value gBC=gCA=value |AB|=|CD|=value |BC|=|DA|=value | Merit point (E) |
Rectangle (Also need to prove it is not a square for E) | 4 sides. All opposite sides are parallel. All opposite sides are same length. (Adjacent sides different length.) | gAB=gCD=value gBC=gCA=value |AB|=|CD|=value |BC|=|DA|=value |AB|=value≠|BC|=value |BC|=value≠|CD|=value | Merit point (E point with full proof showing its not a square.) |
Definitions: Quadrilaterals 2
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Shape | Definition | Suggested evidence (without the numbers) | A point: Shapes with one defining property. M point: Shapes with two defining properties.. |
Square | 4 sides. All sides have same length. All opposite sides same gradient. (E: to prove not rhombus, show that no adjacent sides are perpendicular.) | |AB|=|BC|=|CD|=|DA|=value gAB=gCD=value gBC=gDA=value gAB×gBC=value×value=-1 gBC×gCD=value×value=-1 | Merit point (E) |
Right angled trapezium (For E, show that it is not a parallelogram) | 4 sides. Only 1 pair of opposite sides are parallel. 2 Adjacent sides are perpendicular. (The pair of parallel sides are of different length.) | |AB|=value≠|BC|=value |BC|=value≠|CD|=value gAB×gBC=value×value=-1 gBC×gCD=value×value≠-1 | Merit point (E) |
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Definitions: Quadrilaterals 3
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Shape | Definition | Suggested evidence (you need to add the values) | A point: Shape with 1 Defining property. M point: With 2 defining properties. |
Rhombus that is not a square | 4 sides. All sides same length. All opposite sides are parallel. (For E point: Adjacent sides not perpendicular.)
| |AB|=|BC|=|CD|=|DA|=value gAB×gBC=value×value≠-1 gBC×gCD=value×value≠-1 gAB=gCD=value gBC=gDA=value | Merit point (E) |
Identifying shapes with reason and evidence
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Identifying shapes with reason and evidence
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Shape:
Reason:
Evidence:
Identifying shapes with reason and evidence
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Shape: Triangle ABC is a right angled triangle.
Reason: It is a right angled triangle because it has 3 sides and one right angle.
Evidence: We know it is a right angle, because the gradient of AC=1 and the gradient of AB=-1 are each others negative reciprocal as 1×(-1)=-1.
Identifying shapes with reason and evidence
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Shape:
Reason:
Evidence:
Identifying shapes with reason and evidence
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Shape: Quadrilateral ABCD is a parallelogram and rectangle.
Reason: It is a parallelogram because it is a quadrilateral with two pairs of parallel sides. It is also a rectangle because it has two pairs of parallel sides, the opposite sides are of equal length and it has 4 right angles.
Evidence: It has 2 pairs of parallel sides because:
grad AB=grad CD=3
grad BC=grad AD=-3
Opposite sides are equal because:
|AB|=|CD|=6.324 units
|AD|=|BC|=3.162 units
It has 4 right angles because:
grad AB×grad BC=3×(-3)=-1
grad BC×grad CD=(-3)×3=-1
grad CD×grad AD=3×(-3)=-1
grad AD×grad AB=-3×(3)=-1
Identifying shapes with reason and evidence
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Shape:
Reason:
Evidence:
Identifying shapes with reason and evidence
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Shape: Quadrilateral ABCD is a trapezium that’s not a parallelogram, rhombus, rectangle, or a square.
Reason: It is a trapezium because it has one pair of opposite sides that are parallel. It is not a rectangle or a square because only two sides are parallel.
Evidence: We know that one pair of opposite sides are parallel as grad AB=grad=CD=0.25.
Good luck (if you need it)!
/The mathematics department
2.1 Coordinate Geometry (AS91256)