Modern Analytic Geometry

William Wooton, Edwin F. Beckenbach, Frank J. Fleming

Chapter 3 Applications of Lines - all with Video Answers

Educators

Section 1

Distance between a Point and a Line

Problem 1

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(3,4) ; \mathbf{v}=(2,1), \mathbf{T}(4,7)$

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Problem 2

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(2,5) ; \mathbf{v}=(3,-1), \mathbf{T}(1,3)$

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Problem 3

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-1,3) ; \mathbf{v}=(2,2), \mathbf{T}(4,-2)$

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Problem 4

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(4,-7) ; \mathbf{v}=(-5,-6), \mathbf{T}(1,0)$

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Problem 5

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-5,1) ; \mathbf{v}=(4,-6), \mathbf{T}(0,-1)$

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Problem 6

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-6,-2) ; \mathbf{v}=(1,5), \mathbf{T}(0,0)$

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Problem 7

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(1,6) ; m=2, \mathbf{T}(3,3)$

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Problem 8

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(5,1) ; m=-3, \mathbf{T}(2,-1)$

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Problem 9

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(4,-2) ; m=\frac{1}{2}, \mathbf{T}(5,-3)$

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Problem 10

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-3,7) ; m=-\frac{2}{3}, \mathbf{T}(-4,6)$

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Problem 11

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-3,-4) ; m=-\frac{1}{7}, \mathbf{T}(2,3)$

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Problem 12

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-4,0) ; m=-4, \mathbf{T}(0,4)$

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Problem 13

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$S(5,7) ; 3 x+4 y+12=0$

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Problem 14

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(3,6) ; 2 x-3 y+6=0$

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01:33

Problem 15

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(-2,4) ; 5 x-4 y-10=0$

Dwijendra Rao

Numerade Educator

Problem 16

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(-3,7) ; 6 x-5 y-15=0$

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12:33

Problem 17

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(4,-3) ; x-8 y+5=0$

Donald Albin

Numerade Educator

12:33

Problem 18

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(5,-8) ; 4 x+y-3=0$

Donald Albin

Numerade Educator

13:28

Problem 19

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(-1,-5) ; 5 x-12 y+7=0$

Donald Albin

Numerade Educator

06:28

Problem 20

find the distance between the point $\mathbf{S}$ and the line $\mathfrak{L}$ with the given equation.
$\mathbf{S}(-3,-6) ; 4 x-3 y+6=0$

Donald Albin

Numerade Educator

00:48

Problem 21

find the lengths of the altitudes of the triangle whose vertices R, S, and T are given.
$\mathbf{R}(0,4), \mathbf{S}(4,-3), \mathbf{T}(-3,1)$

Vicki Stebbins

Numerade Educator

02:20

Problem 22

find the lengths of the altitudes of the triangle whose vertices R, S, and T are given.
$\mathbf{R}(1,0), \mathbf{S}(2,5), \mathbf{T}(-2,2)$

Jay Patel

Numerade Educator

00:48

Problem 23

find the lengths of the altitudes of the triangle whose vertices R, S, and T are given.
$\mathbf{R}(7,0), \mathbf{S}(-1,0), \mathbf{T}(1,-1)$

Vicki Stebbins

Numerade Educator

00:48

Problem 24

find the lengths of the altitudes of the triangle whose vertices R, S, and T are given.
$\mathbf{R}(4,-1), \mathbf{S}(1,7), \mathbf{T}(-3,3)$

Vicki Stebbins

Numerade Educator

02:06

Problem 25

Find the area of the triangle whose vertices $\mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ are given in Exercises $21-24$.

James Kiss

Numerade Educator

02:06

Problem 26

Find the area of the triangle whose vertices $\mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ are given in Exercises $21-24$.

James Kiss

Numerade Educator

02:06

Problem 27

Find the area of the triangle whose vertices $\mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ are given in Exercises $21-24$.

James Kiss

Numerade Educator

02:06

Problem 28

Find the area of the triangle whose vertices $\mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ are given in Exercises $21-24$.

James Kiss

Numerade Educator

02:13

Problem 29

find the distance between the parallel lines with equations as given.
$3 x-y-8=0$ and $3 x-y-15=0$

Erin Kearney

Numerade Educator

01:27

Problem 30

find the distance between the parallel lines with equations as given.
$x-3 y+12=0$ and $x-3 y-18=0$

Erin Kearney

Numerade Educator

08:46

Problem 31

Find $k$ so that the point $(2, k)$ is equidistant from the lines with equations $x+y-2=0$ and $x-7 y+2=0$.

Asif Khan

Numerade Educator

02:54

Problem 32

Find $k$ so that the point $(k, 4)$ is equidistant from the lines with equations $13 x-9 y-10=0$ and $x+3 y-6=0$.

Laurie Huffman

Numerade Educator

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Problem 33

find equations of the bisectors of the angles whose sides lie in the lines with the given equations.
$3 x+4 y-2=0$ and $4 x+3 y+2=0$

Victor Salazar

Numerade Educator

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Problem 34

find equations of the bisectors of the angles whose sides lie in the lines with the given equations.
$3 x-4 y+1=0$ and $5 x+12 y-2=0$

Victor Salazar

Numerade Educator

Problem 35

find equations of the bisectors of the angles whose sides lie in the lines with the given equations.
$x+3 y-2=0$ and $2 x-6 y+5=0$

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Problem 36

find equations of the bisectors of the angles whose sides lie in the lines with the given equations.
$x+y-6=0$ and $3 x-3 y+5=0$

Victor Salazar

Numerade Educator

06:19

Problem 37

Find equations of the angle bisectors of the triangle with vertices $\mathbf{R}(6,2)$, $\mathbf{S}(-2,-4)$, and $\mathbf{T}\left(-\frac{42}{5}, 8\right)$. (Hint: Use a sketch to help select the required equations.)

Avi Zellman

Numerade Educator

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Problem 38

Find equations of the angle bisectors of the triangle whose sides lie in the lines with equations $x+2 y-4=0, x-2 y+2=0$, and $2 x-y-8=0$.

Victor Salazar

Numerade Educator

01:44

Problem 39

Find equations of the lines that are parallel to the line $\&$ with equation $3 x-4 y+10=0$ and are located a distance of 5 units from $\&$.

Jacquelyn Trost

Numerade Educator

01:44

Problem 40

Find equations of the lines that are parallel to the line $\&$ with equation $15 x+8 y-34=0$ and are located a distance of 4 units from $£$.

Jacquelyn Trost

Numerade Educator

01:32

Problem 41

Find the distance between the point $\mathbf{S}(b, a)$ and the line with $x$ - and $y$-intercepts $a$ and $b$, respectively.

Adrian Co

Numerade Educator

01:16

Problem 42

Find the distance between the point $\mathbf{S}(b,-a)$ and the line with $x$ - and $y$-intercepts $a$ and $b$, respectively.

Erika Bustos

Numerade Educator

05:26

Problem 43

Show that the area of the triangle with vertices $\mathbf{R}\left(x_1, y_1\right), \mathbf{S}\left(x_2, y_2\right)$, and $\mathbf{T}\left(x_3, y_3\right)$ is

$$
\frac{1}{2}\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|
$$

Vikash Ranjan

Numerade Educator