Please enable JavaScript.
Coggle requires JavaScript to display documents.
ANALYTYIC GEOMETRY - Coggle Diagram
ANALYTYIC GEOMETRY
LINE EQUATIONS
Equation of a Line in Slope - Point Form: Let A(x₁,y₁) is a point on the line ℓ and slope of the line is m, the equation is
-
Equation of the lines which are passing through the origin can be found by y = mx (m is gradient of line) formula.
Equation of a Line in Two Point Form: Equation of the line with two points A(x₁,y₁) and B(x₂,y₂) on the line is,
-
-
-
Equation of a line parallel to y-axis: md is not defined since tan90º is not defined
x-a = 0 since md = y-b / x-a is not defined
x= a
Equation of a line passing through origin: The line d passing through the point (0,0) has slope
m = y-0 / x-0
y = mx
LINES IN PLANE
Slope (Gradient) Of a Straight Line: The slope of a straight line shows how steep a straight line is. The method to calculate the slope is: Divide the change in height by the change in horizontal distance
-
-
-
-
Slope of a Line Passing Through Two Points: The slope of a line passing through the points A(x₁,y₁) and B(x₂,y₂) is
-
Parallel and Perpendicular Lines: d₁ and d₂ are two lines, m₁ and m₂ are the slopes of the lines.
-
-
Don't Forget: If the line is parallel to the x-axis then m = 0 (zero slope) If the line is parallel to the y-axis then m is undefined
Collinear points have same slopes. If the points A,B and C are collinear points, then mAB = mAC = mBC
THE COORDINATE PLANE
We use the notation (x,y) to represent an ordered pair, notice that the x-coordinate or abscissa is first, followed by the y-coordinate or ordinate listed second. To represent an ordered pair graphically, we use the rectangular coordinate system also called the Cartesian Coordinate System named after the French Mathematician Rene Descartes.
The plane used in this system is called the Coordinate Plane or Cartesian plane. The horizontal axis (x-axis) and the vertical axis (the y-axis) intersect at the origin 0 and naturally divide the Cartesian plane into 4 quadrants labeled quadrants 1,2,3 and 4.
Plotting Ordered Pairs in the Cartesian Plane: To plot the point (-2,3), go 2 units to the left of the origin on the x-axis then move 3 units up parallel to the y-axis. The point corresponding to the ordered pair (-2,3) is labeled in the figure below and is located in Quadrant 2.
If abscissa of a point is zero, the point lies on y-axis,
If ordinate of a point is zero, the point lies on x-axis
Finding the Distance between Two Points using the Distance Formula: We can use the Pythagorean Theorem to find the distance between any two points in a plane. To find the length d(A,B) of the line segment AB, consider the point C(x₁,y₂) which is on the same vertical line segments as point A and the same horizontal line segment as point B.
The triangle formed by points A,B and C is a right triangle whose hypotenuse has length d(A,B). The horizontal leg of the triangle has length a = |x₂-x₁| while the vertical leg of the triangle has length b = |y₂-y₁|
The Distance Formula: The distance between any two points A(x₁,y₁) and B(x₂,y₂) is given by the formula. d(A,B) = √(x₂-x₁)² + (y₂-y₁)²
Dividing Line Segment Into Proportional Parts: If C ∈ [AB], (internal division). The coordinates of a point C that divides the line segment joining the points A(x₁,y₁) and B(x₂,y₂) in the ratio of |CA| / |CB| = k are:
-
If C ∉ |AB| , (external division). The coordinated of a point C that divides [AB] in the ratio of |CA| / |CB| = k are:
-
Straight Line Graphs: To be able to draw a straight line, we need to know at least two different points that the line passes through. After the points are plotted on Cartesian coordinate plane, draw a straight line passing on the points
If the equation of the line is given ax + by = 0 form or y = mx form, the graph of line passes on origin.
If the equation of the line is given by + c = 0 form or y = b form, the graph of line is parallel to x-axis.
If the equation of the line is given ax + c = 0 form or x = a form, the graph of line is parallel to y-axis.
-
THE NUMBER LINE
A real number line, or simply number line, allows us to visually display real numbers by associating them with unique points on a line. There are infinitely many real numbers between any two distinct given real numbers. The real number associated with a point is called a coordinate. A(x) means that x is the coordinate of the point A.
The Distance Between Two Points: The distance between two points A(x) and B(y) on the number line is the absolute value of the coordinated of these points.
-
-
Finding the Midpoint of a Line Segment using the Midpoint Formula: Suppose we wish to find the midpoint C(x₀,y₀) of the line segment from A(x₁,y₁) and B(x₂,y₂). To find this midpoint, we simply "average" the x and y coordinates respectively. In other words, the x coordinate of the midpoint is x₁+x₂ / 2 while the y- coordinate of the midpoint is y₁+y₂ / 2.
Midpoint Formula: The midpoint of the line segment from A(x₁,y₁) to B(x₂,y₂) is
(x₁ + x₂ / 2 , y₁+y₂ / 2)
Note: Let A,B,C,D are the vertices of a parallelogram ABCD and K be the intersection of diagonals.
-
Center of Gravity of a Triangle: Let A(x₁,y₁) , B(x₂,y₂) and C(x₃,y₃) are the vertices of the triangle ABC
G = (x₁+x₂+x₃ / 3 , y₁+y₂+y₃ / 3)
Finding the Slope of the Line with a Given Equation: ax + by + c = 0 is linear equation of a line, y = mx + n is functional form a line. The slope of a line given in the form y = mx + n is m. To find the slope of a line given in the form of ax + by + c = 0, we have the convert the equation to the form y = mx +n.
-