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The ______ of a nonhorizontal line is the positive angle $\theta$ (less than $\pi )$ measured counterclockwise from the $x$ -axis to the line.

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If a nonvertical line has inclination $\theta$ and slope $m,$ then $m=$ ______.

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If two nonperpendicular lines have slopes $m_{1}$ and $m_{2},$ then the angle between the two lines is $\tan \theta=$ ______.

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The distance between the point $\left(x_{1}, y_{1}\right)$ and the line $A x+B y+C=0$ is given by $d=$ ______.

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Find the slope of the line with inclination $\theta$.

$\theta=\frac{\pi}{6}$ radian

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Find the slope of the line with inclination $\theta$.

$\theta=\frac{\pi}{4}$ radian

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Find the slope of the line with inclination $\theta$.

$\theta=\frac{3 \pi}{4}$ radians

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Find the slope of the line with inclination $\theta$.

$\theta=\frac{2 \pi}{3}$ radians

Ethan D.

Numerade Educator

Find the slope of the line with inclination $\theta$.

$\theta=\frac{\pi}{3}$ radians

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Find the slope of the line with inclination $\theta$.

$\theta=\frac{5 \pi}{6}$ radians

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Find the slope of the line with inclination $\theta$.

$\theta=0.26$ radian

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Find the slope of the line with inclination $\theta$.

$\theta=0.74$ radian

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Find the slope of the line with inclination $\theta$.

$\theta=1.27$ radians

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Find the slope of the line with inclination $\theta$.

$\theta=1.35$ radians

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Find the slope of the line with inclination $\theta$.

$\theta=1.81$ radians

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Find the slope of the line with inclination $\theta$.

$\theta=2.88$ radians

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=-1$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=-2$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=1$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=2$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=\frac{3}{4}$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=\frac{1}{2}$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=-\frac{5}{2}$

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Find the inclination $\theta$ (in radians and degrees) of the line with slope $m .$

$m=-\frac{7}{9}$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(\sqrt{3}, 2),(0,1)$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(1,2 \sqrt{3}),(0, \sqrt{3})$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(-\sqrt{3},-1),(0,-2)$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(3, \sqrt{3}),(6,-2 \sqrt{3})$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(6,1),(10,8)$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(12,8),(-4,-3)$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(-2,20),(10,0)$

Karen B.

Numerade Educator

Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$(0,100),(50,0)$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$\left(\frac{1}{4}, \frac{3}{2}\right),\left(\frac{1}{3}, \frac{1}{2}\right)$

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Find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$\left(\frac{2}{5},-\frac{3}{4}\right),\left(-\frac{11}{10},-\frac{1}{4}\right)$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$2 x+2 y-5=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$x-\sqrt{3} y+1=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$3 x-3 y+1=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$\sqrt{3} x-y+2=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$x+\sqrt{3} y+2=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$-2 \sqrt{3} x-2 y=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$6 x-2 y+8=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$2 x-6 y-12=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$4 x+5 y-9=0$

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Find the inclination $\theta$ (in radians and degrees) of the line.

$5 x+3 y=0$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$3 x+y=3$

$x-y=2$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$x+3 y=2$

$x-2 y=-3$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$\begin{aligned} x-y &=0 \\ 3 x-2 y &=-1 \end{aligned}$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$2 x-y=2$

$4 x+3 y=24$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$x-2 y=7$

$6 x+2 y=5$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$5 x+2 y=16$

$3 x-5 y=-1$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$x+2 y=8$

$x-2 y=2$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$3 x-5 y=3$

$3 x+5 y=12$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$0.05 x-0.03 y=0.21$

$0.07 x+0.02 y=0.16$

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Find the angle $\theta$ (in radians and degrees) between the lines.

$0.02 x-0.05 y=-0.19$

$0.03 x+0.04 y \quad =0.52$

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Find the slope of each side of the triangle, and use the slopes to find the measures of the interior angles.

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Find the distance between the point and the line.

Point $\quad$ Line

$(1,1) \quad y=x+1$

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Find the distance between the point and the line.

Point $\quad$ Line

$(2,1) \quad y=x+2$

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Find the distance between the point and the line.

Point $\quad$ Line

$(3,2) \quad y=2 x-1$

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Find the distance between the point and the line.

Point $\quad$ Line

$(1,4) \quad y=4 x+2$

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Find the distance between the point and the line.

Point $\quad$ Line

$(-2,6) \quad y=-x+5$

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Find the distance between the point and the line.

Point $\quad$ Line

$(4,-4) \quad y=-2 x-3$

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Find the distance between the point and the line.

Point $\quad$ Line

$(1,-3) \quad y=2 x-5$

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Find the distance between the point and the line.

Point $\quad$ Line

$(-2,8) \quad y=-3 x+2$

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Find the distance between the point and the line.

Point $\quad$ Line

$(2,3) \quad 3 x+y=1$

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Find the distance between the point and the line.

Point $\quad$ Line

$(2,1) \quad-2 x+y=2$

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Find the distance between the point and the line.

Point $\quad$ Line

$(6,2) \quad-3 x+4 y=-5$

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Find the distance between the point and the line.

Point $\quad$ Line

$(1,-3) \quad 4 x-3 y=-7$

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Find the distance between the point and the line.

Point $\quad$ Line

$(-1,2) \quad 5 x+3 y=-4$

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Find the distance between the point and the line.

Point $\quad$ Line

$(2,-3) \quad 4 x-5 y=-2$

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Find the distance between the point and the line.

Point $\quad$ Line

$(-1,-5) \quad 6 x+3 y=3$

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Find the distance between the point and the line.

Point $\quad$ Line

$(-5,-3) \quad-2 x-6 y=7$

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The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.

$A(-1,0), B(0,3), C(3,1)$

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The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.

$A(-4,0), B(0,5), C(3,3)$

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The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.

$A(-3,0), B(0,-2), C(2,3)$

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The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.

$A(-2,0), B(0,-3), C(5,1)$

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The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.

$A(1,1), B(2,4), C(3,5)$

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The points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C,$ and $(c)$ find the area of the triangle.

$A(-3,-2), B(-1,-4), C(3,-1)$

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A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

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A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

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A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof.

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A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel.

(a) Draw a diagram that gives a visual representation of the problem.

(b) Find the inclination of the conveyor.

(c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

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Find the angles $\alpha$ and $\beta$ shown in the drawing of the roof truss.

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The Falls Incline Railway in Niagara Falls, Ontario, Canada, is an inclined railway

that was designed to carry people from the City of Niagara Falls to Queen Victoria Park. The railway is approximately 170 feet long with a 36$\%$ uphill grade (see figure).

(a) Find the inclination $\theta$ of the railway.

(b) Find the change in elevation from the base to the top of the railway.

(c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track.

(d) Sketch a graph of the equation you found in part (c).

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Determine whether the statement is true or false. Justify your answer.

A line that has an inclination greater than $\pi / 2$ radians has a negative slope.

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Determine whether the statement is true or false. Justify your answer.

To find the angle between two lines whose angles of inclination $\theta_{1}$ and $\theta_{2}$ are known, substitute $\theta_{1}$ and $\theta_{2}$ for $m_{1}$ and $m_{2},$ respectively, in the formula for the angle between two lines.

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Determine whether the statement is true or false. Justify your answer.

The inclination of a line is the angle between the line and the $x$ -axis.

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Use the pentagon shown below.

(a) Describe how you can use the formula for the distance between a point and a line to find the area of the pentagon.

(b) Describe how you can use the formula for the angle between two lines to find the measures of the interior angles of the pentagon.

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Consider a line with slope $m$ and $y$ -intercept $(0,4) .$

(a) Write the distance $d$ between the origin and the line as a function of $m .$

(b) Graph the function in part (a).

(c) Find the slope that yields the maximum distance between the origin and the line.

(d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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Consider a line with slope $m$ and $y$ -intercept $(0,4) .$

(a) Write the distance $d$ between the point $(3,1)$ and the line as a function of $m .$

(b) Graph the function in part (a).

(c) Find the slope that yields the maximum distance between the point and the line.

(d) Is it possible for the distance to be 0$?$ If so, what is the slope of the line that yields a distance of 0$?$

(e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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Explain why the inclination of a line can be an angle that is greater than $\pi / 2$ , but the angle between two lines cannot be greater than $\pi / 2$ .

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