Podcast
Questions and Answers
______ Geometry, also known as analytic geometry, describes the relationship between geometry and algebra using graphs that include curves and lines.
______ Geometry, also known as analytic geometry, describes the relationship between geometry and algebra using graphs that include curves and lines.
Coordinate
The position of points on a plane, in coordinate geometry, is represented by an ______ of numbers.
The position of points on a plane, in coordinate geometry, is represented by an ______ of numbers.
ordered pair
The Cartesian coordinate system is divided into four ______ by two axes that are perpendicular.
The Cartesian coordinate system is divided into four ______ by two axes that are perpendicular.
quadrants
The point on the coordinate plane where the x- and y-axes cross is known as the ______.
The point on the coordinate plane where the x- and y-axes cross is known as the ______.
A pair of numbers (x, y) that describe a point's location on a plane are known as its ______.
A pair of numbers (x, y) that describe a point's location on a plane are known as its ______.
The study of geometry using coordinate points is known as either coordinate geometry or ______ geometry.
The study of geometry using coordinate points is known as either coordinate geometry or ______ geometry.
Coordinate geometry allows calculation of the distance between two points, the midpoint of a line, and the ______ of a triangle in the Cartesian plane.
Coordinate geometry allows calculation of the distance between two points, the midpoint of a line, and the ______ of a triangle in the Cartesian plane.
In the distance formula, the distance d between points A and B is calculated as $d = \sqrt{(x_2-x_1)^2 + (y_2-______)^2}$
In the distance formula, the distance d between points A and B is calculated as $d = \sqrt{(x_2-x_1)^2 + (y_2-______)^2}$
The section formula is used to find a point which divides a line into a given ______.
The section formula is used to find a point which divides a line into a given ______.
If $l = k$, the point C dividing a line is a ______ with the formula $C = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$.
If $l = k$, the point C dividing a line is a ______ with the formula $C = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$.
To find an area when given four coordinates in the Cartesian plane, one can use a formula based on ______.
To find an area when given four coordinates in the Cartesian plane, one can use a formula based on ______.
The formula to find the ______ of a line joining two points, A and B is given by $m = \frac{y_2 - y_1}{x_2 - x_1} = tan \theta$.
The formula to find the ______ of a line joining two points, A and B is given by $m = \frac{y_2 - y_1}{x_2 - x_1} = tan \theta$.
The slope, represented by m, indicates a change in y along the line for a ______ change in x.
The slope, represented by m, indicates a change in y along the line for a ______ change in x.
The slope of a line may also be known as its ______.
The slope of a line may also be known as its ______.
Instead of directly using the slope formula, the slope can be calculated by understanding and working it out ______.
Instead of directly using the slope formula, the slope can be calculated by understanding and working it out ______.
The "run" in slope calculation is the ______ distance between the left and right points of a line.
The "run" in slope calculation is the ______ distance between the left and right points of a line.
The slope of the line can also be expressed as an ______, usually in degrees or radians.
The slope of the line can also be expressed as an ______, usually in degrees or radians.
A line can have a slope that is positive, negative ______ or undefined.
A line can have a slope that is positive, negative ______ or undefined.
The intercept of a line is the ______ of the point that crosses the y-axis.
The intercept of a line is the ______ of the point that crosses the y-axis.
On the coordinate plane, a ______ line is a straight line where every point has the same y-coordinate.
On the coordinate plane, a ______ line is a straight line where every point has the same y-coordinate.
A horizontal line is parallel with the ______ of the coordinate plane.
A horizontal line is parallel with the ______ of the coordinate plane.
The slope of a horizontal line is ______.
The slope of a horizontal line is ______.
A vertical line is a line on the coordinate plane where all points on the line have the same ______.
A vertical line is a line on the coordinate plane where all points on the line have the same ______.
A vertical line has ______ slope.
A vertical line has ______ slope.
A straight line on the coordinate plane can be described by the equation $y = mx + c$, this is known as the ______ form.
A straight line on the coordinate plane can be described by the equation $y = mx + c$, this is known as the ______ form.
In the equation $y = mx + c$, c represents the ______ where the line crosses the y-axis.
In the equation $y = mx + c$, c represents the ______ where the line crosses the y-axis.
Using the point slope form, the equation of a line with slope m that passes through (x1, y1) is $y - y_1 = m(x - ______)$
Using the point slope form, the equation of a line with slope m that passes through (x1, y1) is $y - y_1 = m(x - ______)$
If two lines do not intersect, they are said to be ______.
If two lines do not intersect, they are said to be ______.
Parallel lines have the same steepness, hence they have the same ______.
Parallel lines have the same steepness, hence they have the same ______.
The ______ is the only distinction between the two parallel lines.
The ______ is the only distinction between the two parallel lines.
The lines are parallel if the ______ are the same and the y-intercepts are different.
The lines are parallel if the ______ are the same and the y-intercepts are different.
Perpendicular lines, unlike parallel lines, do ______.
Perpendicular lines, unlike parallel lines, do ______.
The intersection of perpendicular lines creates a ______ degree angle.
The intersection of perpendicular lines creates a ______ degree angle.
The slope of one line is equal to the ______ of the slope of the other line, if two lines are perpendicular.
The slope of one line is equal to the ______ of the slope of the other line, if two lines are perpendicular.
Perpendicular line slopes differ from one another in a ______ way.
Perpendicular line slopes differ from one another in a ______ way.
The perpendicular distance between two objects is the distance from one to the other measured along the line that is ______ to one or both.
The perpendicular distance between two objects is the distance from one to the other measured along the line that is ______ to one or both.
When two lines intersect one of the two pairs of lines the intersection generates will be considered the ______ angle and the other is obtuse.
When two lines intersect one of the two pairs of lines the intersection generates will be considered the ______ angle and the other is obtuse.
When two lines intersect, the angle between the lines is defined as the smallest of these angles or the ______ angle.
When two lines intersect, the angle between the lines is defined as the smallest of these angles or the ______ angle.
The angle between two lines is denoted by $\theta$ and is defined as $\theta = tan^{-1} \frac{m_1-______}{1+m_1m_2}$
The angle between two lines is denoted by $\theta$ and is defined as $\theta = tan^{-1} \frac{m_1-______}{1+m_1m_2}$
The equations of the ______ of the angles are derived by considering the distances from a point to the two lines forming the angles.
The equations of the ______ of the angles are derived by considering the distances from a point to the two lines forming the angles.
Flashcards
Coordinate Geometry
Coordinate Geometry
A branch of geometry in which the position of points on a plane is represented by an ordered pair of numbers.
Cartesian coordinate system
Cartesian coordinate system
A number line called the cartesian plane that is divided into four quadrants by two axes.
Origin
Origin
The point where the x- and y-axes cross on the coordinate plane.
Coordinates
Coordinates
Signup and view all the flashcards
Coordinate geometry
Coordinate geometry
Signup and view all the flashcards
Distance formula
Distance formula
Signup and view all the flashcards
Section formula
Section formula
Signup and view all the flashcards
Formula of Area
Formula of Area
Signup and view all the flashcards
Slope formula
Slope formula
Signup and view all the flashcards
Slope of a line
Slope of a line
Signup and view all the flashcards
Intercept of a line
Intercept of a line
Signup and view all the flashcards
Equation of horizontal line
Equation of horizontal line
Signup and view all the flashcards
Equation of vertical line
Equation of vertical line
Signup and view all the flashcards
Slope-intercept form
Slope-intercept form
Signup and view all the flashcards
Point-slope form
Point-slope form
Signup and view all the flashcards
Equation of a line through 2 points
Equation of a line through 2 points
Signup and view all the flashcards
Parallel lines
Parallel lines
Signup and view all the flashcards
Perpendicular lines
Perpendicular lines
Signup and view all the flashcards
Perpendicular distance
Perpendicular distance
Signup and view all the flashcards
Angle between two straight lines
Angle between two straight lines
Signup and view all the flashcards
Study Notes
Coordinate Geometry
- Coordinate geometry, or analytic geometry, links geometry and algebra via graphs with curves and lines.
- It applies algebraic techniques to solve geometric problems.
- Points on a plane are shown as ordered number pairs.
Cartesian Coordinate System
- This system uses the Cartesian plane, divided into quadrants by two perpendicular axes.
- The horizontal axis is termed the x-axis, while the vertical one is the y-axis.
- The origin is located where the x and y axes cross.
- Coordinates (x, y) show a point's position on the plane.
- Coordinates in the Cartesian plane are visualized as (3, 4).
- The distance between two points and the midpoint of the interval connecting them can be calculated with coordinate knowledge.
- The study of geometry via coordinate points is termed coordinate or analytic geometry.
- Coordinate geometry helps calculate the distance between two points, the midpoint of a line, m:n ratio of a line and the area of a triangle in the Cartesian plane.
Distance Formula
- The distance (d) between points A and B is calculated as: d = √((x₂ - x₁)² + (y₂ - y₁)²)
Section Formula
- To find the intersection point dividing it into a l:k ratio
- If l ≠ k, coordinate C = ((lx₂ + kx₁) / (l + k), (ly₂ + ky₁) / (l + k)).
- If l = k, coordinate C is the midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2).
Area Formula
-
To find an area when given four coordinates in the Cartesian plane A (x₁,y₁), B (x₂,y₂), C (x₃,y₃), and D (x₄,y₄)
-
Triangle ABC Area = 1/2 |x₁y₂ + x₂y₃ + x₃y₁ - x₂y₁ + x₃y₂ + x₁y₃|
-
Rectangle ABCD area = 1/2 |(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)|
Slope Formula
- the slope (m) of a line joining points A and B is found using: m = (y₂ - y₁) / (x₂ - x₁) = tan θ
Slope and Intercept of a Line
- A line's slope measures its steepness represented by 'm', showing the change in 'y' for a unit change in 'x'.
- Known as the gradient, the slope indicates how steep a line is.
Calculating Slope
- Find the "run" (dx), or horizontal distance = x₂ - x₁.
- Find the "rise" (dy), or vertical distance = y₂ - y₁.
- Divide the rise by the run: slope = dy / dx.
- Slope is also expressed as an angle: θ = tan⁻¹(m), with m = tan(θ).
Slope Direction
- Positive: Line slopes upwards from left to right.
- Negative: Line slopes downwards from left to right.
- Zero: Horizontal line (no rise or fall).
- Undefined: Vertical line.
Intercept of a Line
- The y-intercept is termed 'b', is where the line crosses the y-axis.
- With slope 'm' and a point (x, y), calculate 'b' using: b = y - mx.
Equation of a Line in the Cartesian Plane
- Horizontal line: y = b, has zero slope, is parallel to the x-axis, and all points have the same y-coordinate.
- Vertical line: x = a, has an undefined slope, is parallel to the y-axis, and all points share the same x-coordinate.
Equation of a Line (Slope-Intercept Form)
- Defined as y = mx = c, uses the slope 'm' and y-intercept 'c'.
Point-Slope Form
- Expresses a line with slope 'm', passing through point (x₁, y₁)
- Defined as: y - y₁ = m(x - x₁)
Equation of a Line Through Two Points
- Given two points (x₁, y₁) and (x₂, y₂), the equation is: (y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁)
Rules of a Line in the Cartesian Plane
- Parallel Lines: Do not intersect, same slope, distinct y-intercepts.
- Perpendicular Lines: Intersect at 90°, slopes are negative reciprocals.
- For lines f(x) = m₁x + b₁ and g(x) = m₂x + b₂:
- Parallel if m₁ = m₂.
- Perpendicular if m₁m₂ = -1.
Perpendicular Distance
- The formula for distance ‘d’ between point P(m, n) and line l: Ax + By + C = 0 is d = |Am + Bn + C| / √(A² + B²).
Angle Between Two Straight Lines
- The smallest between two intersection lines is calculated as θ = tan⁻¹|(m₁ - m₂) / (1 + m₁m₂)| where m₁ and m₂ are the slopes.
Equation of Bisectors
- For lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the bisectors are given by: (a₁x + b₁y + c₁) / √(a₁² + b₁²) = ± (a₂x + b₂y + c₂) / √(a₂² + b₂²)
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
