Lines

1
Lines

• Geometry, Summer School 2007

2
Parallel Lines

• Line a straight path extending without end in
  two directions.
• Parallel two lines are parallel if they do not
  intersect (ever!) and if they are coplanar
• When writing about lines, we use the
  symbol.
• When drawing parallel lines, we use
  triangle/arrow symbols.

3
Perpendicular Lines

• Perpendicular two lines are perpendicular if
  they intersect to form a 90 angle.
• When writing about perpendicular lines, use the
  ? symbol.
• When drawing perpendicular lines, use the square
  symbol.
• (If two lines intersect but do not have the
  symbol, do not assume that they are
  perpendicular!)

4
Parallel Perpendicular Lines

• If there is a line and a point not on the line,
  then there is exactly one line through the
  point parallel to the given line.
• If there is a line and a point not on the line,
  then there is exactly one line through the
  point perpendicular to the given line.

5
Skew Lines

• Skew lines are lines that do not intersect and
  are not coplanar. They only exist in 3-D!
• Careful! When drawing skew lines, it may look
  like they intersect, but they dont!

6
Activity

• Parallel, Perpendicular, and Skew Lines in 3-D!
• Use your candy box and your spaghetti to answer
  these questions on a piece of paper.

7
Slope

• The slope of a line is the steepness of a line.
  More specifically, it is the ratio of vertical
  change (rise) to horizontal change (run) between
  any two points on the line.

8
Positive Negative Slope

• Lines with a positiveslope rise from left
  toright.
• Lines with a negative slope fall from left to
  right.

9
Horizontal Vertical Lines

• Horizontal lines have zero slope. They have a
  y-value.
• Vertical lines haveundefined slope. Theyhave
  an x-value.

10
Slope of Lines Examples

• Find the slope of all the lines shown.

11
Slopes of Parallel Lines

• If two lines are parallel, what can you say about
  their slopes???

12
Slopes of Perpendicular Lines

• If two lines are perpendicular, their slopes are
  negative reciprocals of each other.
• Also, the product of their slopes is -1.

13
Line Example

• Find the slope of each line. Which lines are
  parallel?

14
Line Example

• Line m passes through (-1,3) and (4,1). Line t
  passes through (-1,-1) and (3,-3). Are the two
  lines parallel? Explain how you know.

15
Line Example

• Line n passes through (0,2) and (6,5). Line m
  passes through (2,4) and (4,0). Is n ? m?
  Explain how you know.

16
Line Example

• Line k passes through (0,3) and (5,2). Graph the
  line perpendicular to k that passes through the
  point (1,2).

17
Your Turn
18
Slope-intercept form

• The slope-intercept form is the general form of a
  line. It is given by the formula
• y mx b

19
Slope-intercept form Example

• Write an equation of the line in slope-intercept
  form.

20
Slope-intercept form Example

• Write an equation of the line passing through the
  point (-1,1) that is parallel to the line with
  the equation y 2x 3.

21
Slope-intercept form Example

• Write an equation of the line j passing through
  the point (2,3) that is perpendicular to the line
  k with the equation y -2x 2.

22
Slope-intercept form Example

• Write the equation of the line that passes
  through (0,-3) and (1,-2).

23
Slope-intercept form Your turn
24
A Real-World Example

• The graph models the total cost of joining a gym.
  Write an equation of the line. Explain the
  meaning of the slope and the y-intercept of the
  line.

25
Standard Form

• Standard form is another way of writing the
  formula for a line. It is in the format Ax By
  C where A and B are not both zero.
• Ex 3x 4y 12.

26
Standard Form Example

• Graph the lines 2x 3y 6y 4x -3

27
Real-Life Example

• You can rent DVDs at a local store for 4 each.
  An Internet company offers a flat fee of 15 a
  month for as many rentals as you want. How many
  DVDs do you need to rent to make the online
  rental a better buy?
• Method 1
• Use a table.

28
Real-Life Example

• Method 2 Model each rental with an equation.
  Graph each equation.

29
Real-Life Example

• Method 3 Model each rental with an equation and
  then solve using algebra.

30
Challenge Problem

• Find a value for k so that the line through
  (-1,k) and (-7,-2) is parallel to the line with
  the equation y x 1.