Lines. We have learned that the graph of a linear equation. y = mx +b

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1 Section 0. Lines We have learne that the graph of a linear equation = m +b is a nonvertical line with slope m an -intercept (0, b). We can also look at the angle that such a line makes with the -ais. This is calle the line s inclination. Definition of Inclination The inclination of a nonhorizontal line is the positive angle θ (less than π) measure counterclockwise from the -ais to the line. θ = 0 Horizontal Line Vertical Line Acute Angle Obtuse Angle

2 Section 0. The inclination of a line is relate to its slope in the following wa: Inclination an Slope If a nonvertical line has inclination θ an slope m, then m = tan θ. Eample: Fin the inclination of the line 5. Solution: The slope of the line is, so for its inclination, tan. Because the slope is positive, we know that the graph goes up an to the right, an thus makes an acute angle with the -ais. This correspons to a Quarant I angle an we can use arctan( ½ ) to fin the answer. arctan 6. 6

3 Eample: Fin the inclination of the line 5. CHAT Pre-Calculus Section 0. Solution: The equation of the line can be written as 5, which tells us that slope of the line is, so for its inclination, tan. Because the slope is negative, we know that the graph goes own an to the right, an thus makes an obtuse angle with the -ais. This correspons to a Quarant II angle. If we use arctan to fin the answer, our calculator tells us arctan( ) 6. 4 This is because the range for the arctangent function is from to. (or -90 to 90 ) Since our angle θ is obtuse (think of a Quarant II angle), we nee to fin θ b using 6.4 as the reference angle

4 Section 0. The Angle Between Two Lines If two istinct lines intersect an are not perpenicular, then their intersection forms two pairs of opposite angles (also calle vertical angles). The smaller of these angles is calle the angle between the two lines. Look at the graph of the intersecting lines. θ θ θ Because θ is the eterior angle of the triangle, it must be equal to the sum of the remote interior angles. Thus, θ + θ = θ So, θ = θ - θ where θ < θ. 4

5 Section 0. Using the formula for the formula of the ifference of two angles, we get tan tan( ) tan tan tan tan This is from the formula for tan(u-v) Since m tan, we en up with the following formula: Angle Between Two Lines If two nonperpenicular lines have slopes m an m, the angle between the two lines is m m tan m m *Note: The reason that we use the absolute value brackets is because the angle we are fining is acute, which means its tangent must be positive. 5

6 Eample: Fin the angle between the lines given b line : + = 8 an line : 4 5 =. Solution: Line : 4 m = Line : 4 4 m = CHAT Pre-Calculus Section 0. Use the formula: tan m m m m tan m m m m So, if tan, then tan.484 raians or 85 6

7 Section 0. The Distance Between a Point an a Line Fining the istance from a point to a line is fining the length of the perpenicular segment that joins the point to the line. (, ) Distance Between a Point an a Line The istance between the point (, ) an the line A + B + C = 0 is A A B B C 7

8 Section 0. Eample: Fin the istance between the point (, ) an the line given b 4 =. Solution: Put the equation in the form A + B + C = = 0 Fin the istance using the formula: A A () B B 4() ( 4) C

9 Section 0. 9 Eample: For the triangle with vertices A(-, ), B(, ), an C(, -), fin a) the altitue, an b) the area of the triangle. a) We will fin the istance from B to the line AC. First fin the equation of the line AC. ) ( m Use this an one of the points in = m + b. ) ( b b b m 0 b m A B C

10 Section 0. Now, using the equation an point B, fin the istance from the point B to the line AC. This is the altitue. A A () B B () C The altitue is. b) To fin the area, we will nee to know the length of AC. To o this, use the points A(-, ) an C(, -) in the istance formula. ( ) ( ) ( ) ( ) ( () ( )) So, AC =. Now fin the area of the triangle. 0

11 Section 0. The formula for the area of a triangle is A bh. A A A bh. ( ) square units.