Section 10.1 Notes

Definition of Inclination

The inclination of a nonhorizontal line is thepositive angle θ (less than π or 180°) measuredcounterclockwise from the x-axis to the line.

y

x

θ = 0π or 0°

Horizontal Line

Vertical Line

or 902

x

y

Obtuse Angle

θx

y

Acute Angle

θ x

y

m

(x1, 0)

(x2, y1)

1

2 1y

x x

tanθ 1

2 1y

x x

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

5x – y + 3 = 0y = 5x + 3 m = 5tan θ = 5θ = 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

If two nonperpendicular lines have slopes m1 and m2. The angle between the two lines is found by

2 1

1 2

tan1

m m

m m

x

y

m2m1

θ

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

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

Line 1: 2x + y = 4Line 2: x – y = 2m1 = -2m2 = 1

θ

2x + y = 4

x – y = 2

x

y

tan

tan 3

θ = 1.249 rad.

1 21 1 2

31

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.

x

y

d

Distance Between a Point and a Line

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

1 1

2 2.

Ax By Cd

A B

Example 3

Find the distance between the point (0, 2) and the line 4x + 3y = 7.

The general form of the equation is4x + 3y – 7 = 0

So, the distance between the point and the line is

d

2 2

4 0 3 2 7

4 3

1 units5

Example 4

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.

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.

1 0 13 0 3

� AC

m

10 03

y x

13

y x

3 0 x y

x

y

AC

B

h

So, the distance between this line and the point (1, 5) is

2 2

1 1 3 5 14 7 10Altitude units5101 3

h

The area of the triangle is12

A bh

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

2 23 0 1 0 10 b

1 1410 7 square units2 10

A