ANALYTIC GEOMETRY
ANALYTIC OF LINES
ANALYTIC OF POINTS
THE CARTESIAN SYSTEM
SLOPE
We use the notation (𝑥,𝑦) to represent the ordered pair of points.
We use the word “quadrant” for each of the four regions into which the plane is divided by.
FINDING THE DISTANCE BETWEEN TWO POINTS
If the point lies on abscissa, point y will be zero. If the point lies on ordinate, point x will be zero.
The Pythagorean Theorem: a² + b² = c²
The Distance Formula: √ [ (x₂ - x₁)² + (y₂ - y₁)² ]
FINDING THE MIDPOINT
Measure the distance between the two end points, and divide the result by 2.
FINDING THE CENTER OF GRAVITY
G(x, y) = (x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3
The Mid-Point of the Diagonals of a Parallelogram: x₁ + x₃ = x₂ + x₄, y₁ + y₃ = y₂ + y₄
Slope (denoted by the letter m) is numerical measure of a line's inclination relative to the horizontal.
Slope is calculated by finding the ratio of the vertical change to the horizontal change between (any) two distinct points on a line. Slope also can be found by using the tangent of the angle of inclination.
For acute angles tanθ value will be positive, for obtuse angles tanθ value will be negative.
LINE EQUATIONS
LINES
Equation of a Line in Slope - Point Form: y - y₁ = m.(x - x₁). Also equation of a line that is passing through the origin can be found by using the y = mx, coefficient (m) of the x is the slope.
Equation of a Line in Two Point Form: y - y₁ / x - x ₁ = y₂ - y₁ / x₂ - x₁
Standard Equation of a Line: ax + by + c = 0
Equation of a Line Parallel to X-Axis: Let P(a,b) is a point that is parallel to x-axis, we can denote the equation by y = b. Equation of a Line Parallel to Y-Axis: Let P(a,b) is a point that is parallel to y-axis, we can denote the equation by x = a.
Equation of a Line With Known Intercepts on the Axes: x / a + y / b = 1.
Equation of a Line Passing Through Origin: y = mx.
Finding the Slope of the Line with a Given Equation: We have to use the y=mx+b form to find the slope. You can easily see the slope since it is the coefficient of the x variable, or the number in front of x.
VERTICAL
HORIZONAL
Because the m = tan90° is undefined, the slope of vertical lines are also undefined.
m = tan0° is equal to 0, so that horizontal lines have a slope of 0 as well.
FINDING THE SLOPE OF A LINE PASSING THROUGH TWO POINTS
m = y₂ - y₁ / x₂ - x₁ (note: we can find the slope of the line passing through the origin with m = y / x).
PARALLEL
PERPENDICULAR
m₁ = m₂
m₁ . m₂= -1
Note: Collinear points have same slopes.
STRAIGHT LINE GRAPHS
For straight line graphs there is a linear relationship between the x and y values. We should know at least 2 different points that the line passes through to draw a straight line.
If the equation is in the y = b form, the line is parallel to x-axis. If the equation is in the x = a form, then the line is parallel to y-axis. Also the form ax + by = 0 gives us the graph of line passes on origin.