Points and Lines and Slopes, Oh My!

1
Points and Lines and Slopes, Oh My!

• What is the relationship between the slopes of
  parallel and perpendicular lines?

2
Modeling Mathematics

• Directions
• Draw and cut out a scalene right triangle from
  the small square piece of graph paper.
• Label the triangle ABC where lt C is the right
  angle.
• Label the sides of the triangle as shown.

3
Modeling Mathematics

• Directions
• Place triangle on coordinate plane where B lies
  on the origin and side a lies along the positive
  x-axis. Fill in the coordinates of B and slope
  of side c in the table under the TRIAL 1 column
  beside the Original Position rows
• Rotate their triangles 90 counterclockwise so
  that B is still at the origin and side a is along
  the positive y-axis. Write the new coordinates
  of B and slope of side c in the table under the
  TRIAL 1 column beside the 90 counterclockwise
  rotation rows.
• Now move the triangle down 2 units, to the right
  3 units, and rotate the triangle 180 along point
  B. Write the new coordinates of B and slope of
  side c in the table under the TRIAL 1 column
  beside the 2 units down, 3 units right and 180
  rotation rows.

4
Modeling Mathematics

• Repeat this process TWO MORE TIMES, selecting a
  different starting place each time--i.e. not do
  not place B on (0,0)
• Write your answers under the TRIAL 2 and TRIAL
  3 columns, respectively.
• Place triangle on coordinate plane where B lies
  on the your selected location and side a lies
  along the positive x-axis. Fill in the
  coordinates of B and slope of side c in the table
  under the appropriate TRIAL column beside the
  Original Position rows
• Rotate their triangles 90 counterclockwise so
  that B is still at your selected location and
  side a is along the positive y-axis. Write the
  new coordinates of B and slope of side c in the
  table under the appropriate TRIAL column beside
  the 90 counterclockwise rotation rows.
• Now move the triangle down 2 units, to the right
  3 units, and rotate the triangle 180 along point
  B. Write the new coordinates of B and slope of
  side c in the table under the appropriate TRIAL
  column beside the 2 units down, 3 units right and
  180 rotation rows.

5
Definitions
Perpendicular Lines Lines that intersect at right
angles are called perpendicular lines If the
product of the slopes of two lines is -1, then
the lines are perpendicular. and the converse
is also true If two lines are perpendicular, then
the product of the slopes is -1.

• Parallel Lines
• Lines in the same plane that never intersect are
  called parallel.
• If two non-vertical lines have the same slope,
  then they are parallel.
• and the converse is also true
• If two non-vertical lines are parallel, then they
  have the same slope.

6
What is the relationship (if any) between the two
lines?
Example 1
Example 2
Example 3
y (1/4)x 11 y 5x 8 2y 3x 2y 4x
-6 y 5x 1 y -3x 2
7
Check for Understanding
Hold up the GREEN card if the lines or pairs of
points are parallel

• Hold up the RED card if the lines or pairs of
  points are perpendicular

8
Parallel or Perpendicular?
9
Perpendicular!
10
Parallel or Perpendicular?
11
Parallel!
12
Parallel or Perpendicular?
13
Parallel!
14
Parallel or Perpendicular?
15
Parallel!
16
Parallel or Perpendicular?
17
Perpendicular!
18
Write an equation in slope-intercept form of the
line that passes through the given point and is
parallel to each equation.
Example 1
Example 2
x 3y 8 2x 3y 6 (5, -4)
(-3, 2)








19
Write an equation in slope-intercept form of the
line that passes through the given point and is
parallel to each equation.
Example 1
Example 2
2x 9y 5 y (1/3)x 2
(6, -13) (-3, 1)








20
Putting it all together
Lines x, y, and z all pass through point (-3,
4). Line x has slope 4 and is perpendicular to
line y. Line z passes through Quadrants I and
II only. (1) Write an equation for each line.
(2) Graph the three lines on the same coordinate
plane.