F18 Week 11 Coordinate Geometry
Slope: Formula for Finding Slope
If 'm' represents the slope of a line and A and B are points with coordinates (x₁, y₁) and (x₂, y₂) respectively, then the slope of the line passing through A and B is given by the following formula. m = y₂ - y₁ / x₂ - y₂, if x₂ ≠ x₁ Copy and paste the following link into your browser to learn more about using the formula for finding slope in geometry: https://youtu.be/JUMpzQxKxJA
Slope: Parallel Lines
If lines are PARALLEL, they slant in exactly the same direction. If they are nonvertical, their steepness is exactly the same. ▶︎ THEOREM 103: If two nonvertical lines are parallel, then they have the same slope. ▶︎ THEOREM 104: If two lines have the same slope, then the lines are nonvertical parallel lines. Copy and paste the following link into your browser to learn more about the slope of parallel and perpendicular lines geometry: https://youtu.be/TqH4UHLMXx4
Distance Formula
DISTANCE FORMULA (also know as Theorem 101) specifies: ➜ If coordinates of two points are (x₁, y₁) and (x₂, y₂), then the distance, d, between the two points is given by the distance formula (see formula details in the attachment). For example, use the Distance Formula to find the distance between the points with coordinates (−3, 4) and (5, 2). Let (-3, 4) = x₁, y₁ and (5, 2) = x₂, y₂, then... d = √(5-[-3])² + (2 - 4)² d = √(8)² + (2)² d = √64 + 4 d = √68 d = √(4)(17) d = 2√17 Copy and paste the following link into your browser to learn about using the distance formula in geometry: https://youtu.be/TRyXoK9EpOU
Equation of Lines: Standard Form
Equations involving one or two variables can be graphed on any x− y coordinate plane. In general, the following principles are true: ➜ If a point lies on the graph of an equation, then its coordinates make the equation a true statement. ➜ If the coordinates of a point make an equation a true statement, then the point lies on the graph of the equation. A LINEAR EQUATION is any equation whose graph is a line. All linear equations can be written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both zero This form for equations of lines is known as the STANDARD FORM for the equation of a line.
Points and Coordinates
Every point in space can be assigned three numbers with respect to a starting point. Those three numbers allow us to distinguish any point from any other point in space. Fortunately for you, we are not dealing here with three dimensions, but only with two. ▶︎ COORDINATES OF A POINT: Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers. ▶︎ X‐AXIS AND Y‐AXIS: To locate points in a plane, two perpendicular lines are used—a horizontal line called the x‐axis and a vertical line called the y‐axis. ▶︎ ORIGIN: The point of intersection of the x‐axis and y‐axis. The coordinates [ordered pair] for the origin are (0, 0). ▶︎ COORDINATE PLANE: The x‐axis, y‐axis, and all the points in the plane they determine. ▶︎ ORDERED PAIRS: Every point in a coordinate plane is named by a pair of numbers whose order is important; these numbers are written in parentheses and separated by a comma . ▶︎X‐COORDINATE : The number to the left of the comma in an ordered pair is the x‐coordinate of the point and indicates the amount of movement along the x‐axis from the origin. The movement is to the right if the number is positive and to the left, if the number is negative. ▶︎ Y‐COORDINATE: The number to the right of the comma in an ordered pair is the y‐coordinate of the point and indicates the amount of movement perpendicular to the x‐axis. The movement is above the x‐axis if the number is positive and below the x‐axis if the number is negative. ▶︎ QUADRANTS: The x‐axis and y‐axis separate the coordinate plane into four regions called quadrants. The upper right quadrant is quadrant 1; the upper left quadrant is quadrant II; the lower left quadrant is quadrant III, and the lower right quadrant is quadrant IV. Copy and paste the following link into your browser to learn more about the role of points and coordinates in coordinate geometry: https://youtu.be/xC3qGv0IMcE
Slope: Perpendicular Lines
If two lines are PERPENDICULAR and neither one is vertical, then one of the lines has a positive slope, and the other has a negative slope. Also, the absolute values of their slopes are reciprocals. ▶︎ THEOREM 105: If two nonvertical lines are perpendicular, then their slopes are opposite reciprocals of one another, or the product of their slopes is −1. ▶︎ THEOREM 106: If the slopes of two lines are opposite reciprocals of one another, or the product of their slopes is −1, then the lines are nonvertical perpendicular lines. Horizontal and vertical lines are always perpendicular: therefore, two lines, one of which has a zero slope and the other an undefined slope are perpendicular Copy and paste the following link into your browser to learn more about the slope of parallel and perpendicular lines geometry: https://youtu.be/TqH4UHLMXx4
Midpoint Formula
Numerically, the midpoint of a segment can be considered to be the AVERAGE OF ITS ENDPOINTS. This concept helps in remembering a formula for finding the midpoint of a segment given the coordinates of its endpoints. Recall that the average of two numbers is found by dividing their sum by two. MIDPOINT FORMULA (also known as Theorem 102): If the coordinates of A and B are (x₁, y₁) and (x₂, y₂), respectively, then the midpoint, M, of AB is given by the following midpoint formula: M = x₁ + x₂ / 2, y₁ + y₂ / 2 Copy and paste the following link into your browser to learn about using the midpoint formula in geometry: https://youtu.be/udtWqcmXt7k
Summary Coordinate Geometry Formulas
Summary of Coordinate Geometry Formulas ▶︎▶︎ STANDARD FORM: Ax + By = C, where A, B, and C are real numbers, also A and B are both NOT zero. ▶︎▶︎ POINT-SLOPE FORM: y - y₁ = m (x - x₁), where (x₁, y₁) is a point on the line and 'm' is the slope of the line. ▶︎▶︎ SLOPE-INTERCEPT FORM: y = mx + b, where 'm' is the slope of the line and 'b' is the y-intercept value.
Slope of a Line: Definition
The SLOPE OF A LINE IS a measurement of the steepness and direction of a nonvertical line. When a line rises from left to right, the slope is a POSITIVE number. When a line falls from left to right, the slope is a NEGATIVE number. The x‐axis or any line parallel to the x‐axis has a slope of ZERO. The y‐axis or any line parallel to the y‐axis has no defined (or UNDEFINED) slope.
Equation of Lines: X-Intercept and Y-Intercept
The X‐INTERCEPT OF A GRAPH is the point where the graph intersects the x‐axis. ➜ It always has a y‐coordinate of zero. ➜ A horizontal line that is not the x‐axis has no x‐intercept . The Y-INTERCEPT OF A GRAPH is the point where the graph intersects the y‐axis. ➜ It always has an x‐coordinate of zero. ➜ A vertical line that is not the y‐axis has no y‐intercept . One way to graph a linear equation is to find solutions by giving a value to one variable and solving the resulting equation for the other variable. A minimum of TWO points is necessary to graph a linear equation. Copy and paste the following link into your browser to learn more about graphing linear equations using x-intercept and y-intercept: https://youtu.be/mxBoni8N70Y