Points and Lines and Points and Lines and Slopes, Slopes, Oh My!Oh My!

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

Modeling MathematicsModeling Mathematics

Directions:• Draw and cut out a

scalene right triangle from the small square piece of graph paper.

• Label the triangle ABC where < C is the right angle.

• Label the sides of the triangle as shown.

Modeling Mathematics 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.

Modeling MathematicsModeling MathematicsRepeat 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.

Definitions Definitions Parallel LinesLines 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.

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.

What is the relationship (if What is the relationship (if any) between the two lines?any) between the two lines?

y = (1/4)x + 11 y = 5x – 8 2y – 3x =2y + 4x = -6 y = 5x + 1 y = -3x + 2

Example 1 Example 2

Example 3

Check for UnderstandingCheck 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

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)

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)

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.