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Warm Up: California Standards Lesson Presentation

This lesson presentation covers warm-up exercises for finding reciprocals, slope of a line, parallel and perpendicular lines, and identifying slopes of lines.

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Warm Up: California Standards Lesson Presentation

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Find the reciprocal. 1. 2 2. 3. Find the slope of the line that passes through each pair of points. 4. (2, 2) and (–1, 3) 5.(3, 4) and (4, 6) 6. (5, 1) and (0, 0) 3 2

  3. Vocabulary parallel lines perpendicular lines

  4. The slopeof a line in a coordinate plane is a Ratio that describes the steepness (Rise over Run) of the line. Any two points on a line can be used to determine the slope.

  5. One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

  6. CD Example 1: Finding the Slope of a Line Use the slope formula to determine the slope the line. Substitute (4, 2) for (x1, y1) and(–2, 1) for (x2, y2) in the slope formula and then simplify.

  7. Use the slope formula to determine the slope of JK through J(3, 1) and K(2, –1). TEACH! Example 1 Substitute (3, 1) for (x1, y1) and (2, –1) for (x2, y2) in the slope formula and then simplify.

  8. WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3) TEACH: Example 1a Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. Vertical and horizontal lines are perpendicular.

  9. BC and DEfor B(1, 1), C(3, 5), D(–2, –6), and E(3, 4) TEACH: Example 1c Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. The lines have the same slope, so they are parallel.

  10. Remember! In a parallelogram, opposite sides are parallel.

  11. Use the ordered pairs and the slope formula to find the slopes of MJ and KL. MJ is parallel to KL because they have the same slope. JK is parallel to ML because they are both horizontal. Example 2: Show that JKLM is a parallelogram. Since opposite sides are parallel, JKLM is a parallelogram.

  12. Use the ordered pairs and slope formula to find the slopes of AD and BC. AD is parallel to BC because they have the same slope. AB is parallel to DC because they are both horizontal. TEACH! Example 2 Show that the points A(0, 2), B(4, 2), C(1, –3), D(–3, –3) are the vertices of a parallelogram. B(4, 2) • A(0, 2) • • • C(1, –3) D(–3, –3) Since opposite sides are parallel, ABCD is a parallelogram.

  13. Perpendicular lines are lines that intersect to form right angles (90°).

  14. Identify which lines are perpendicular: y = 3; x = –2; y = 3x; . y = 3 x = –2 y =3x Example 3: Identifying Perpendicular Lines The graph given by y = 3 is a horizontal line, and the graph given by x = –2 is a vertical line. These lines are perpendicular.

  15. y = 3 x = –2 The slope of the line given by y = 3x is 3. The slope of the line described by y =3x is . Continue These lines are perpendicular because the product of their slopes is –1.

  16. Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = x = 3 The slope of the line described by y – 6 = 5(x + 4) is 5. The slope of the line described by y = is y = –4 y – 6 = 5(x + 4) TEACH! Example 3 The graph described by x = 3 is a vertical line, and the graph described by y = –4 is a horizontal line. These lines are perpendicular.

  17. Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = x = 3 y = –4 y – 6 = 5(x + 4) TEACH! Example 3 Continued These lines are perpendicular because the product of their slopes is –1.

  18. Slopes of Parallel and Perpendicular Lines • The slopes of two nonvertical lines are equal. • Two lines with the same slope are parallel • Vertical lines are parallel • The product of the slopes of two perpendicularlines, neither of which is vertical, is -1. • If the product of the slopes of two lines is -1, then the two lines are perpendicular • A horizontal line and a vertical line are perpendicular.

  19. Helpful Hint If you know the slope of a line, the slope of a perpendicular line will be the "opposite reciprocal.”

  20. Facts: Theorems • Two lines parallel to a third line are parallel to each other. • In a plane, two lines perpendicular to a third line are parallel to each other.

  21. If ABC is a right triangle, AB will be perpendicular to AC. slope of slope of AB is perpendicular to AC because Example 4: Show that ABC is a right triangle. Therefore, ABC is a right triangle because it contains a right angle.

  22. Q(2, 6) P(1, 4) slope of PQ R(7, 1) slope of PR PQ is perpendicular to PR because the product of their slopes is –1. TEACH! Example 4 Show that P(1, 4), Q(2, 6), and R(7, 1) are the vertices of a right triangle. If PQR is a right triangle, PQ will be perpendicular to PR. Therefore, PQR is a right triangle because it contains a right angle.

  23. Lesson Quiz: Part I Write an equation in slope-intercept form for the line described. 1. contains the point (8, –12) and is parallel to 2. contains the point (4, –3) and is perpendicular to y = 4x + 5

  24. slope of = XY slope of YZ = 4 slope of = WZ slope of XW = 4 The product of the slopes of adjacent sides is –1. Therefore, all angles are right angles, and WXYZ is a rectangle. Lesson Quiz: Part II 3. Show that WXYZ is a rectangle.

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