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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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.
θ = 0π or 0°
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?
y = 5x + 3
m = 5
tan θ= 5
θ = 1.373 rad.
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.
If two nonperpendicular lines have slopes
m1 and m2. The angle between the two lines is found by
The tan θ must be positive since θ is always an acute angle thus the reason for the absolute value sign in the formula.
2x + y = 4
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.
The distance between the point (x1, y1) and the line Ax + By + C = 0 is found by
The general form of the equation is
4x + 3y – 7 = 0
So, the distance between the point and the line is
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, 5) is
The area of the triangle is