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Understanding the Inclination and Slope of Nonhorizontal Lines in Geometry

micah avatar
micah
25 diapositives · 508 Views · Jul 31, 2014

This section delves into the concepts of inclination and slope for nonhorizontal lines. The inclination is defined as the positive angle θ, measured counterclockwise from the x-axis. We explore how to determine the slope (m) using the tangent of the inclination (m = tan θ). The signs of slopes for acute and obtuse angles are discussed, revealing that acute angles yield positive slopes, whereas obtuse angles yield negative slopes. Additionally, the angle between two intersecting lines and the distance from a point to a line using perpendiculars is analyzed with examples and formulas provided for calculation.

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Understanding the Inclination and Slope of Nonhorizontal Lines in Geometry
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  1. Section 10.1 Notes

  2. Definition of Inclination The inclination of a nonhorizontal line is the positive angle θ (less than π or 180°) measured counterclockwise from the x-axis to the line.

  3. Horizontal Line y θ = 0π or 0° x

  4. Vertical Line y x

  5. Obtuse Angle y θ x

  6. Acute Angle y (x2, y1) θ x (x1, 0)

  7. Inclination and Slope If a nonvertical line has inclination θ and slope m, then m = tan θ

  8. Look at the graphs of an acute angle and an obtuse angle. What is the sign of the slope of the line that has an angle of inclination that is acute? What is the sign of the slope of the line that has an angle of inclination that is obtuse? • The sign of the slope of the line that has an angle of inclination that is acute is always positive. This means that the tangent of an acute angle is always positive. • The sign of the slope of the line that has an angle of inclination that is obtuse is always negative. This means that the tangent of an obtuse angle is always negative.

  9. Example 1 • Graph and find the inclination of the line given by 5x – y + 3 = 0 to the nearest thousandth of a radian.

  10. 5x – y + 3 = 0 y = 5x + 3 m = 5 tan θ= 5 θ = 1.373 rad. θ

  11. Two distinct lines in a plane are either parallel or intersecting. If they intersect and are not perpendicular, their intersection forms two pairs of vertical angles. One pair is acute and the other pair is obtuse. The smaller of these angles is the angle between the two lines.

  12. Angle Between Two Lines If two nonperpendicular lines have slopes m1 and m2. The angle between the two lines is found by

  13. The tan θ must be positive since θ is always an acute angle thus the reason for the absolute value sign in the formula. y θ m2 m1 x

  14. Example 2 • Graph and find the angle between the following two lines to the nearest thousandth of a radian. • Line 1: 2x + y = 4 • Line 2: x – y = 2

  15. Line 1: 2x + y = 4 • Line 2: x – y = 2 • m1 = -2 • m2 = 1 2x + y = 4 θ 3 x – y = 2 θ = 1.249 rad.

  16. Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point and the line.

  17. y d x

  18. Distance Between a Point and a Line The distance between the point (x1, y1) and the line Ax + By + C = 0 is found by

  19. Example 3

  20. Find the distance between the point (0, 2) and the line 4x + 3y = 7. The general form of the equation is 4x + 3y – 7 = 0 So, the distance between the point and the line is

  21. Example 4

  22. Consider a triangle with vertices A(0, 0), B(1, 5), and C(3, 1).a. Find the altitude from vertex B to side AC. b. Find the area of the triangle.

  23. a. To find the altitude, use the formula for the distance between line AC and the point B(1, 5). Find the equation of line AC.

  24. So, the distance between this line and the point (1, 5) is The area of the triangle is

  25. b. Use the distance formula to find the base AC.