Section 10 1 notes
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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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Section 10.1 Notes

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Section 10 1 notes

Section 10.1 Notes


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.


Section 10 1 notes

Horizontal Line

y

θ = 0π or 0°

x


Vertical line

Vertical Line

y

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


Section 10 1 notes

  • 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

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


Section 10 1 notes

5x – y + 3 = 0

y = 5x + 3

m = 5

tan θ= 5

θ = 1.373 rad.

θ


Section 10 1 notes

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

If two nonperpendicular lines have slopes

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


Section 10 1 notes

  • 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


Example 2

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


Section 10 1 notes

  • 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.


Section 10 1 notes

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.


Section 10 1 notes

y

d

x


Distance between a point and a line

Distance Between a Point and a Line

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


Example 3

Example 3


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 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


Section 10 1 notes

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.


Section 10 1 notes

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.


Section 10 1 notes

So, the distance between this line and the point

(1, 5) is

The area of the triangle is


Section 10 1 notes

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


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