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

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

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**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.

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**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.

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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.

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**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.

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**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

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**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

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Parallel or Perpendicular?

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Perpendicular!

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Parallel or Perpendicular?

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Parallel!

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Parallel or Perpendicular?

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Parallel!

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Parallel or Perpendicular?

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Parallel!

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Parallel or Perpendicular?

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Perpendicular!

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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)

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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)

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**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.

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