Points and Lines and Slopes, Oh My!

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Points and Lines and Slopes, Oh My!
What is the relationship between the slopes of parallel and perpendicular lines?

Modeling 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 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 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.

Definitions Perpendicular Lines Parallel 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.

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 =2 y + 4x = -6 y = 5x + 1 y = -3x + 2

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

Parallel or Perpendicular?

Perpendicular!

Parallel or Perpendicular?

Parallel!

Parallel or Perpendicular?

Parallel!

Parallel or Perpendicular?

Parallel!

Parallel or Perpendicular?

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 = x – 3y = (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 = y =(1/3)x (6, -13) (-3, 1)

Line x has slope 4 and is perpendicular to line y.
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.