Podcast
Questions and Answers
What is the term for the point where the axes of a Cartesian plane intersect?
What is the term for the point where the axes of a Cartesian plane intersect?
- Origin (correct)
- Coordinate
- Intersection
- Quadrant
The Cartesian plane is divided into how many quadrants?
The Cartesian plane is divided into how many quadrants?
- Three
- Four (correct)
- Two
- Eight
What represents the position of a point in the number plane?
What represents the position of a point in the number plane?
- A linear relation
- The relation
- The origin
- An ordered pair of numbers (correct)
What is a 'relation' defined as in coordinate geometry?
What is a 'relation' defined as in coordinate geometry?
What is a 'linear relation'?
What is a 'linear relation'?
How do you find the midpoint of a line segment on a coordinate plane?
How do you find the midpoint of a line segment on a coordinate plane?
Which theorem is applied to find the distance between two points on a coordinate plane?
Which theorem is applied to find the distance between two points on a coordinate plane?
What does the gradient of a straight line represent?
What does the gradient of a straight line represent?
How is the gradient (m) of a line segment AB defined, given points A(x1, y1) and B(x2, y2)?
How is the gradient (m) of a line segment AB defined, given points A(x1, y1) and B(x2, y2)?
If a line slopes upwards from left to right, what can be said about its gradient?
If a line slopes upwards from left to right, what can be said about its gradient?
Given $A(x_1, y_1)$ and $B(x_2, y_2)$, which formula correctly calculates the midpoint $P$ of the line segment $AB$?
Given $A(x_1, y_1)$ and $B(x_2, y_2)$, which formula correctly calculates the midpoint $P$ of the line segment $AB$?
What is the gradient of a horizontal line?
What is the gradient of a horizontal line?
What can be said about the gradient of a vertical line?
What can be said about the gradient of a vertical line?
If the gradient of a line is $m = tan(\theta)$, and $\theta$ is the angle the line makes with the positive direction of the x-axis, what does $\theta$ represent?
If the gradient of a line is $m = tan(\theta)$, and $\theta$ is the angle the line makes with the positive direction of the x-axis, what does $\theta$ represent?
For a line with a negative gradient, what type of angle does it form with the positive direction of the x-axis?
For a line with a negative gradient, what type of angle does it form with the positive direction of the x-axis?
Which of the following is the gradient-intercept form of a straight line equation?
Which of the following is the gradient-intercept form of a straight line equation?
In the equation y = mx + c, what does 'c' represent?
In the equation y = mx + c, what does 'c' represent?
What is the 'point-gradient' form of the equation of a straight line, given a point $(x_1, y_1)$ and a gradient $m$?
What is the 'point-gradient' form of the equation of a straight line, given a point $(x_1, y_1)$ and a gradient $m$?
Given the equation of a straight line in intercept form as $x/a + y/b = 1$, what do 'a' and 'b' represent?
Given the equation of a straight line in intercept form as $x/a + y/b = 1$, what do 'a' and 'b' represent?
How can you determine the equation of a straight line if you know two points on the line?
How can you determine the equation of a straight line if you know two points on the line?
What is the general form of the equation of a straight line?
What is the general form of the equation of a straight line?
What is the primary method for sketching a straight line given its equation?
What is the primary method for sketching a straight line given its equation?
When two non-vertical lines are parallel, what is true about their gradients?
When two non-vertical lines are parallel, what is true about their gradients?
If two lines are perpendicular and neither is horizontal or vertical, how are their gradients related?
If two lines are perpendicular and neither is horizontal or vertical, how are their gradients related?
What is meant by a 'family of straight lines'?
What is meant by a 'family of straight lines'?
What term describes the variable that, when changed, modifies the characteristics of a family of lines?
What term describes the variable that, when changed, modifies the characteristics of a family of lines?
What does the graphical solution to a system of two simultaneous linear equations represent?
What does the graphical solution to a system of two simultaneous linear equations represent?
What describes a system of two linear equations that has no solution graphically?
What describes a system of two linear equations that has no solution graphically?
What does it mean if a system of two linear equations has infinitely many solutions?
What does it mean if a system of two linear equations has infinitely many solutions?
What condition must be met for two non-vertical lines to be considered parallel?
What condition must be met for two non-vertical lines to be considered parallel?
Given two lines $y = m_1x + c_1$ and $y = m_2x + c_2$, what condition must be met for these lines to be perpendicular, assuming neither line is horizontal or vertical?
Given two lines $y = m_1x + c_1$ and $y = m_2x + c_2$, what condition must be met for these lines to be perpendicular, assuming neither line is horizontal or vertical?
How do you determine if a point $(x, y)$ lies on a given line?
How do you determine if a point $(x, y)$ lies on a given line?
To find the tangent of the angle of slope for a line, what information is needed?
To find the tangent of the angle of slope for a line, what information is needed?
What is the essential characteristic of the 'general form' of a linear equation?
What is the essential characteristic of the 'general form' of a linear equation?
In modeling real-world scenarios with linear equations, what does the gradient of the line typically represent?
In modeling real-world scenarios with linear equations, what does the gradient of the line typically represent?
The coordinates of point A in the Cartesian plane are (a,b). After the coordinate system is translated such that the origin is now located at (-h,-k), what are the new coordinates of point A?
The coordinates of point A in the Cartesian plane are (a,b). After the coordinate system is translated such that the origin is now located at (-h,-k), what are the new coordinates of point A?
A line is defined by the equation $ax + by + c = 0$. If 'a' is very close to zero, and 'b' is significantly larger, what geometric conclusion can most accurately be made about the line?
A line is defined by the equation $ax + by + c = 0$. If 'a' is very close to zero, and 'b' is significantly larger, what geometric conclusion can most accurately be made about the line?
Flashcards
Number plane
Number plane
The coordinate plane divided into four regions by two perpendicular axes.
Origin
Origin
The point where the axes intersect in the coordinate plane (0,0).
Ordered pair
Ordered pair
A pair of numbers (x, y) representing a point's position in the coordinate plane.
Coordinates
Coordinates
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Relation
Relation
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Linear Relation
Linear Relation
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Midpoint
Midpoint
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Distance
Distance
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Gradient
Gradient
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Perpendicular Lines
Perpendicular Lines
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Parallel lines
Parallel lines
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Angle of slope
Angle of slope
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Gradient-intercept form
Gradient-intercept form
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Point-gradient form
Point-gradient form
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Intercept form
Intercept form
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Horizontal line
Horizontal line
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Vertical line
Vertical line
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General form
General form
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Parallel lines
Parallel lines
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Perpendicular
Perpendicular
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Parallelism
Parallelism
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Parameter.
Parameter.
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Geometry of Simultaneous equations
Geometry of Simultaneous equations
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Study Notes
- Coordinate geometry is reviewed
- Includes finding midpoints, distances, gradients, slope angles, line equations, and simultaneous equations
Objectives of Coordinate Geometry
- Find the midpoint of a line segment
- Find the distance between two points
- Determine the gradient of a straight line.
- Calculate the angle of slope of a straight line using its gradient.
- Use and interpret different forms of a straight line's equation.
- Determine parallelism and perpendicularity conditions for lines.
- Use a parameter to represent families of straight lines.
- Use linear relations to solve problems.
- Solve and use simultaneous linear equations.
Number Plane (Cartesian Plane)
- Divided into four quadrants by two perpendicular axes
- Axes intersect at the origin
- Points represented by ordered pairs (x, y), called coordinates
- Using coordinates, one can find straight line equations, distances, and midpoints
- Coordinate geometry provides foundation for calculus
Relations
- Defined as ordered pairs (x, y)
- Relations can be described with a rule relating x and y values, such as y = 2x + 1
- A relation can be shown graphically on a set of axes
- A relation that creates a straight line graph is a linear relation
Finding the Midpoint of a Line Segment
- Special case: segment parallel to an axis aids in getting general result
- In the general case the coordinates of midpoint P, a line segment AB joining A(x1, y1) and B(x2, y2) are given by:
- P = ((x₁+x₂)/2, (y₁+y₂)/2)
- This follows from triangles APC and PBD being congruent (AAS)
- x coordinate of midpoint will be the average of the x coordinates of the end points
- y coordinate of midpoint will be the average of the y coordinates of the end points
Finding Distance Between Two Points
- For points A(x₁, y₁) and B(x₂, y₂), distance found using Pythagoras, by constructing triangle ABC
- The square of the length of AB = ((x₂ - x₁)² + (y₂ - y₁)²)
- AB = √((x₂ - x₁)² + (y₂ - y₁)²)
Lines and Gradient
- Gradient is often defined as "rise over run"
- Gradient Symbol: m
- Gradient = rise/run = (y₂ - y₁) / (x₂ - x₁)
- Order doesn't matter: (y₂ - y₁) / (x₂ - x₁) = (y₁ - y₂) / (x₁ - x₂)
- Positive gradient -> line slopes upwards from left to right
- Negative gradient -> line slopes downwards from left to right
- Horizontal line -> gradient of zero ( m = 0 )
- Vertical line -> Gradient is undefined
Angle of Slope
- For positive gradients the line forms an acute angle θ with the x-axis
- m = tan θ
- For negative gradients, the line forms an acute angle α with the negative direction of the x-axis, and an obtuse angle θ with the positive x-axis direction. Value of θ will be between 90 and 180 degrees.
- m = -tan α = tan θ
Equation of a Straight Line
- Gradient-intercept form: y = mx + c
Vertical lines
- If horizontal, gradient is 0 and simple y = c, where c is the y-intercept.
- If vertical then gradient is undefined and x = a, where a is the x-intercept.
A Point and a Gradient
- Point-gradient form is y - y₁ = m(x - x₁)
- Given point(x₁, y₁) is located on the line
Two Points
- Use the formula m = (y₂ - y₁) / (x₂ - x₁)
- Then use y − y1 = m(x − x₁)
Intercept Form
- x/a + y/b = 1
- a and b are the x and y intercepts
General Form
- Has form mx + ny + p = 0 (m and n not both 0)
- Describes all straight lines
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