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Section 10.1 Notes. 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. Horizontal Line. y. θ = 0 π or 0°. x. Vertical Line. y. x. Obtuse Angle. y. θ. x.

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Definition of inclination
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.


Horizontal Line

y

θ = 0π or 0°

x



Obtuse angle
Obtuse Angle

y

θ

x


Acute angle
Acute Angle

y

(x2, y1)

θ

x

(x1, 0)


Inclination and slope
Inclination and Slope

If a nonvertical line has inclination θ and

slope m, then

m = tan θ


  • 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.


Example 1
Example 1 What is the

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


5 What is the x – y + 3 = 0

y = 5x + 3

m = 5

tan θ= 5

θ = 1.373 rad.

θ


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.


Angle between two lines
Angle Between Two Lines 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

If two nonperpendicular lines have slopes

m1 and m2. The angle between the two lines is found by


  • The tan 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 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


Example 2
Example 2 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

  • 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


  • Line 1: 2 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 x + y = 4

  • Line 2: x – y = 2

  • m1 = -2

  • m2 = 1

2x + y = 4

θ

3

x – y = 2

θ = 1.249 rad.


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.


y line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point and the line.

d

x


Distance between a point and a line
Distance Between a Point and a Line line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point and the line.

The distance between the point (x1, y1) and the line Ax + By + C = 0 is found by


Example 3
Example 3 line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point and the line.


Find the distance between the point 0 2 and the line 4 x 3 y 7
Find the distance between the point (0, 2) and the line line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point 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


Example 4
Example 4 line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point and the line.


Consider a triangle with vertices line is an application of perpendicular lines. This distance is defined as the length of the perpendicular segment joining the point and the line.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.


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.


So, the distance between this line and the point between line

(1, 5) is

The area of the triangle is



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