Upcoming SlideShare
×

# Lines

430 views

Published on

analytic geometry of lines

• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

### Lines

1. 1. WHAT IS A LINE? • From geometry, we already know that a line is determined by two distinct points; and that it is composed of infinitely many points. • Since a line is a set of points, technically, it is also a locus. • A line is a locus of points that have a constant slope. • A line intersects the x and y axes at the points which are called the x- intercept and y-intercept, respectively.
2. 2. PROPERTIES OF A LINE • constant slope (m) • infinitely many points. • x - intercept • y - intercept y - intercept x - intercept constant slope constant slope constant slope y - intercept x - intercept
3. 3. SLOPE OF A LINE • The slope of a line is determined by the ratio of the difference of the y- coordinates to the x-coordinates of two points on the line. • It is represented as m and describes the steepness of the line it represents. m = 0 m > 0 m < 0 m = undefined • also tells if lines are parallel, perpendicular, or simply intersects. slopes are equal slopes are negative reciprocals slopes are not equal nor negative reciprocals
4. 4. • An intercept of a line is the point that the line intersects with the x or y axis. • The point on the x-axis that a line intersects is the x-intercept while the point on the y-axis that it intersects is the y-intercept. • An x-intercept is written as (a, 0) while a y-intercept is written as (0, b), for any real number a and b. • A vertical line does not have a y-intercept, while a horizontal line does not have an x-intercept. X AND Y INTERCEPTS
5. 5. • A line is represented by an equation of the form Ax + By + C = 0, where A, B, and C are integers. • The properties of a line such as the slope, intercepts, or any pair of points can create the equation of a line. • There are four main forms of the equation of a line, all of which are based on the mentioned properties. EQUATION OF A LINE Two-point Form Point-slope Form Slope-intercept FormIntercepts Form
6. 6. TWO-POINT FORM **Should be written in Ax + By + C = 0 form.
7. 7. POINT-SLOPE FORM
8. 8. INTERCEPTS FORM NOTE: Two-point form can be applied if given the two intercepts.
9. 9. SLOPE - INTERCEPT FORM
10. 10. USING THE SLOPE-INTERCEPT FORM GIVEN SLOPE BUT NOT Y-INTERCEPT We don’t have b, but we know values for x and y. Solve for b, then substitute it back to the equation.
11. 11. GRAPH OF A LINE • Just plot the points. • Connect the points with a straight line. • This is also the approach when the x and y intercepts are given.
12. 12. GRAPH OF A LINE • Just plot the point. • Get a second point by using the slope. (in this example, the slope tells us that another point is two units down and three units to the right of the given point) •Connect the points with a straight line. • This is also done when the slope and an x or y intercept is given.
13. 13. GRAPH OF A LINE
14. 14. SOME NOT-SO-BASIC SITUATIONS INVOLVING LINES • Remember how slopes of parallel and perpendicular lines are related. This is used in determining equations of lines through a point given a line parallel or perpendicular to it. PARALLEL AND PERPENDICULAR LINES slopes are equal slopes are negative reciprocals
15. 15. SOME NOT-SO-BASIC SITUATIONS INVOLVING LINES The slope of the given line will be the slope of the line being asked for because they are parallel. Since we have a slope and a point for the line being asked for, we can use point-slope form.
16. 16. SOME NOT-SO-BASIC SITUATIONS INVOLVING LINES The negative reciprocal of the slope of the given line will be the slope of the line being asked for because they are perpendicular. Since we have a slope and a y- intercept for the line being asked for, we can use slope- intercept form.
17. 17. LINE FAMILIES