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Modern Analytic Geometry

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

Chapter 2

Lines in the Plane - all with Video Answers

Educators

Section 1

Lines and Line Segments in the Plane

Problem 1

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(2,1), \mathbf{T}(0,0)$

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Problem 2

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(3,2), \mathbf{T}(1,1)$

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Problem 3

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(4,-2), \mathbf{T}(4,3)$

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Problem 4

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(5,-6), \mathbf{T}(2,-6)$

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Problem 5

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(-7,2), \mathbf{T}(-3,-1)$

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Problem 6

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(-3,1), \mathbf{T}(4,-2)$

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Problem 7

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(-6,-3), \mathbf{T}(-4,-2)$

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Problem 8

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(-1,-7), \mathbf{T}(-7,-1)$

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Problem 9

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(a, b), \mathbf{T}(b, a)$

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Problem 10

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(2 a, b), \mathbf{T}(3 a, 2 b)$

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Problem 11

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(5,-1), \mathbf{T}(-4,2)$

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00:50

Problem 12

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(6,-2), \mathbf{T}(1,7)$

Yujie Wang
Yujie Wang
College of San Mateo
02:10

Problem 13

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(-3,-5), \mathbf{T}(3,10)$

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator

Problem 14

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(4,7), \mathbf{T}(-5,3)$

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02:17

Problem 15

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(2,5), \mathbf{T}(-10,-1)$

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
02:10

Problem 16

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(-5,3), \mathbf{T}(7,21)$

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator

Problem 17

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathbf{S}(-3,7), \mathbf{T}(4,1)$

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Problem 18

find the coordinates of (a) the midpoint and (b) the points of trisection of the segment whose endpoints $\mathbf{S}$ and T are given.
$\mathrm{S}(5,-2), \mathrm{T}(12,-5)$

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Problem 19

find a parametric vector equation for the segment joining:
$\mathbf{R}(2,5)$ and the midpoint of the segment with endpoints $\mathbf{S}(5,1)$ and $\mathbf{T}(7,-3)$.

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00:53

Problem 20

find a parametric vector equation for the segment joining:
$\mathbf{R}(-2,6)$ and the midpoint of the segment with endpoints $\mathbf{S}(0,3)$ and $\mathbf{T}(4,0)$.

James Kiss
James Kiss
Numerade Educator
02:42

Problem 21

find a parametric vector equation for the segment joining:
The midpoint of the segment with endpoints $\mathbf{Q}(-5,2)$ and $\mathbf{R}(1,6)$ and the point one-third of the way from $\mathbf{S}(-2,6)$ to $\mathbf{T}(1,9)$.

Khushbu Rani
Khushbu Rani
Numerade Educator
01:55

Problem 22

find a parametric vector equation for the segment joining:
The point two-thirds of the way from $\mathbf{Q}(8,-2)$ to $\mathbf{R}(2,7)$ and the point one-fourth of the way from $\mathbf{S}(1,6)$ to $\mathbf{T}(9,10)$.

Monica Miller
Monica Miller
Numerade Educator
04:35

Problem 23

Show that the coordinates $\left(x_0, y_0\right)$ of the midpoint of the segment with endpoints $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given by

$$
x_0=\frac{x_1+x_2}{2}, \quad y_0=\frac{y_1+y_2}{2}
$$

Willis James
Willis James
Numerade Educator
03:56

Problem 24

Show that the coordinates $\left(x^{\prime}, y^{\prime}\right)$ and $\left(x^{\prime \prime}, y^{\prime \prime}\right)$ of the points of trisection of the segment with endpoints $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ are given by

$$
x^{\prime}=\frac{2 x_1+x_2}{3}, \quad y^{\prime}=\frac{2 y_1+y_2}{3}
$$

and

$$
x^{\prime \prime}=\frac{x_1+2 x_2}{3}, \quad y^{\prime \prime}=\frac{y_1+2 y_2}{3}
$$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
00:20

Problem 25

Show that the medians of the triangle with vertices $\mathbf{R}(6,1), \mathbf{S}(-2,3)$, and $\mathbf{T}(2,-7)$ meet in a common point located two-thirds of the way from each vertex to the opposite side. [Hint: Determine the coordinates of the point located two-thirds of the way from each vertex to the midpoint of the opposite side.]

Amrita Bhasin
Amrita Bhasin
Numerade Educator

Problem 26

Repeat Exercise 25 for the triangle with vertices $\mathbf{O}(0,0), \mathbf{S}\left(x_1, y_1\right)$, and $\mathbf{T}\left(x_2, y_2\right)$

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02:58

Problem 27

Prove that if the line $£$ is the graph of the linear Cartesian equation $A x+B y+C=0, A^2+B^2 \neq 0$, then all vectors having geometric representations with initial point and terminal point on $\mathfrak{L}$ are parallel. [Hint: Let $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ be any two points on $\mathfrak{L}$. Show that $A\left(x_2-x_1\right)+B\left(y_2-y_1\right)=0$, and accordingly (Section 1-6) that the vector $\left(x_2-x_1, y_2-y_1\right)$ is perpendicular to the vector $(A, B)$.]

Tom Greenwood
Tom Greenwood
Numerade Educator
02:12

Problem 28

Prove that if the line $\mathfrak{L}$ is the graph of the linear Cartesian equation $A x+B y+C=0, A^2+B^2 \neq 0$, and $\mathbf{v}$ is a vector having a geometric representation with one endpoint on $£$ and the other not on $\mathcal{L}$, then $\mathbf{v}$ is not parallel to any nonzero vector having a geometric representation with both endpoints on \&. [Hint: Let $\mathbf{S}\left(x_1, y_1\right)$ be any point on $£$ and $\mathbf{T}\left(x_2, y_2\right)$ be any point not on $£$. Show that $A\left(x_2-x_1\right)+B\left(y_2-y_1\right) \neq 0$, and accordingly that the vector $\left(x_2-x_1, y_2-y_1\right)$ is not perpendicular to the vector $(A, B)$.]

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator