Section 1
The Rectangular Coordinate System
Plot and then label the points $A(0,-3), B(3,-4), C(5,6)$ $D(-2,-5),$ and $E(-3,5)$
Give the coordinates of each point $A, B, C, D,$ and $E .$ Also name the quadrant in which each point lies.(GRAPH CANNOT COPY)
Find the distance between each pair of points:a) $\quad(5,-3)$ and $(5,1)$b) $\quad(-3,4)$ and $(5,4)$c) $(0,2)$ and $(0,-3)$d) $(-2,0)$ and $(7,0)$
If the distance between $(-2,3)$ and $(-2, a)$ is 5 units, find all possible values of $a$
If the distance between $(b, 3)$ and $(7,3)$ is 3.5 units, find all possible values of $b$
Find an expression for the distance between $(a, b)$ and $(a, c)$ if $b>c$
Find the distance between each pair of points:a) $(0,-3)$ and $(4,0)$b) $(-2,5)$ and $(4,-3)$c) $(3,2)$ and $(5,-2)$d) $\quad(a, 0)$ and $(0, b)$
Find the distance between each pair of points:a) $(-3,-7)$ and $(2,5)$b) $\quad(0,0)$ and $(-2,6)$c) $(-a,-b)$ and $(a, b)$d) $(2 a, 2 b)$ and $(2 c, 2 d)$
Find the midpoint of the line segment that joins each pair of points:a) $(0,-3)$ and $(4,0)$b) $(-2,5)$ and $(4,-3)$c) $(3,2)$ and $(5,-2)$d) $(a, 0)$ and $(0, b)$
Find the midpoint of the line segment that joins each pair of points:a) $\quad(-3,-7)$ and $(2,5)$b) $\quad(0,0)$ and $(-2,6)$c) $(-a,-b)$ and $(a, b)$d) $(2 a, 2 b)$ and $(2 c, 2 d)$
Points $A$ and $B$ have symmetry with respect to the origin $O .$ Find the coordinates of $B$ if $A$ is the point:a) $\quad(3,-4)$b) $\quad(0,2)$c) $(a, 0)$d) $\quad(b, c)$
Points $A$ and $B$ have symmetry with respect to point $C(2,3)$ Find the coordinates of $B$ if $A$ is the point:a) $\quad(3,-4)$b) $\quad(0,2)$c) $(5,0)$d) $\quad(a, b)$
Points $A$ and $B$ have symmetry with respect to point $C .$ Find the coordinates of $C$ given the points:a) $A(3,-4)$ and $B(5,-1)$b) $\quad A(0,2)$ and $B(0,6)$c) $A(5,-3)$ and $B(2,1)$d) $A(2 a, 0)$ and $B(0,2 b)$
Points $A$ and $B$ have symmetry with respect to the $x$ axis. Find the coordinates of $B$ if $A$ is the point:a) $\quad(3,-4)$b) $\quad(0,2)$c) $(0, a)$d) $\quad(b, c)$
Points $A$ and $B$ have symmetry with respect to the $x$ axis. Find the coordinates of $A$ if $B$ is the point:a) $\quad(5,1)$b) $\quad(0,5)$c) $(-6, a)$d) $\quad(b, c)$
Points $A$ and $B$ have symmetry with respect to the vertical line where $x=2 .$ Find the coordinates of $A$ if $B$ is the point:a) $\quad(5,1)$b) $\quad(0,5)$c) $(2, a)$d) $(b, c)$
Points $A$ and $B$ have symmetry with respect to the vertical line where $x=2 .$ Find the coordinates of $A$ if $B$ is the point:a) $(5,1)$b) $\quad(0,5)$c) $(-6, a)$d) $\quad(b, c)$
Points $A$ and $B$ have symmetry with respect to the $y$ axis. Find the coordinates of $A$ if $B$ is the point:a) $\quad(3,-4)$b) $\quad(2,0)$c) $(a, 0)$d) $\quad(b, c)$
Points $A$ and $B$ have symmetry with respect to either the $x$ axis or the $y$ axis. Name the axis of symmetry for:a) $A(3,-4)$ and $B(3,4)$b) $A(2,0)$ and $B(-2,0)$c) $A(3,-4)$ and $B(-3,-4)$d) $A(a, b)$ and $B(a,-b)$
Points $A$ and $B$ have symmetry with respect to a vertical line $(x=a)$ or a horizontal line $(y=b) .$ Give an equation such as $x=3$ for the axis of symmetry for:a) $A(3,-4)$ and $B(5,-4)$b) $\quad A(a, 0)$ and $B(a,-b)$c) $A(7,-4)$ and $B(-3,-4)$d) $A(a, 7)$ and $B(a,-1)$
Apply the Midpoint Formula.$M(3,-4)$ is the midpoint of $\overline{A B},$ in which $A$ is the point $(-5,7) .$ Find the coordinates of $B$
Apply the Midpoint Formula.$M(2.1,-5.7)$ is the midpoint of $\overline{A B},$ in which $A$ is the point $(1.7,2.3) .$ Find the coordinates of $B$
Apply the Midpoint Formula.A circle has its center at the point $(-2,3) .$ If one endpoint of a diameter is at $(3,-5),$ find the other endpoint of the diameter.
Apply the Midpoint Formula.A rectangle $A B C D$ has three of its vertices at $A(2,-1)$ $B(6,-1),$ and $C(6,3) .$ Find the fourth vertex $D$ and the area of rectangle $A B C D$
A rectangle $M N P Q$ has three of its vertices at $M(0,0)$ $N(a, 0),$ and $Q(0, b) .$ Find the fourth vertex $P$ and the area of the rectangle $M N P Q$.
Use the Distance Formula to determine the type of triangle that has these vertices:a) $A(0,0), B(4,0),$ and $C(2,5)$b) $\quad D(0,0), E(4,0),$ and $F(2,2 \sqrt{3})$c) $G(-5,2), H(-2,6),$ and $K(2,3)$
Use the method of Example 4 to find the equation of the line that describes all points equidistant from the points $(-3,4)$ and $(3,2)$
Use the method of Example 4 to find the equation of the line that describes all points equidistant from the points $(1,2)$ and $(4,5)$
For coplanar points $A, B,$ and $C,$ suppose that you have used the Distance Formula to show that $A B=5, B C=10,$ and $A C=15 .$ What can you conclude regarding points $A, B$ and $C ?$
If two vertices of an equilateral triangle are at $(0,0)$ and $(2 a, 0),$ what point is the third vertex?(GRAPH CANNOT COPY)
The rectangle whose vertices are $A(0,0), B(a, 0), C(a, b)$ and $D(0, b)$ is shown. Use the Distance Formula to draw a conclusion concerning the lengths of the diagonals $\overline{A C}$ and $\overline{B D}$
There are two points on the $y$ axis that are located a distance of 6 units from the point $(3,1)$ Determine the coordinates of each point.
There are two points on the $x$ axis that are located a distance of 6 units from the point $(3,1) .$ Determine the coordinates of each point.
The triangle that has vertices at $M(-4,0), N(3,-1)$ and $Q(2,4)$ has been boxed in as shown. Find the area of $\triangle M N Q$(GRAPH CANNOT COPY)
Use the method suggested in Exercise 33 to find the area of $\triangle R S T,$ with $R(-2,4), S(-1,-2),$ and $T(6,5)$
Determine the area of $\triangle A B C$ if $A=(2,1), B=(5,3)$ and $C$ is the reflection of $B$ across the $x$ axis.
Find the area of $\triangle A B C$ in Exercise $35,$ but assume that $C$ is the reflection of $B$ across the $y$ axis.
Refer to formulas for Chanter 9Find the exact volume of the solid that results when the triangular region with vertices at $(0,0),(5,0),$ and $(0,9)$ is rotated about thea) $x$ axis.b) $y$ axis.
Refer to formulas for Chanter 9Find the exact volume of the solid that results when the triangular region with vertices at $(0,0),(6,0),$ and $(6,4)$ is rotated about thea) $x$ axis.b) $y$ axis.
Refer to formulas for Chanter 9Find the exact volume of the solid formed when the rectangular region with vertices at $(0,0),(6,0),(6,4)$ and $(0,4)$ is revolved about thea) $x$ axis.b) $y$ axis.
Refer to formulas for Chanter 9Find the exact volume of the solid formed when the region bounded in Quadrant I by the axes and the lines $x=9$ and $y=5$ is revolved about thea) $x$ axis.b) $y$ axis.
Find the exact lateral area of each solid in Exercise $40 .$
Refer to formulas for Chanter 9Find the volume of the solid formed when the triangular region having vertices at $(2,0)$$(4,0),$ and $(2,4)$ is rotated about the $y$ axis.(FIGURE CANNOT COPY)
By definition, an ellipse is the locus of points whose sum of distances from two fixed points $F_{1}$ and $F_{2}$ (called foci) is constant. In the grid provided, find points whose sum of distances from points $F_{1}(3,0)$ and $F_{2}(-3,0)$ is $10 .$ That is, locate some points for which $P F_{1}+P F_{2}=10 ;$ point $P(5,0)$ is one such point. Then sketch the ellipse. (GRAPH CANNOT COPY)
By definition, a hyperbola is the locus of points whose positive difference of distances from two fixed points $F_{1}$ and $F_{2}$ (called foci) is constant. In the grid provided, find points whose difference of distances from points $F_{1}(5,0)$ and $F_{2}(-5,0)$ is $6 .$ That is, locate some points for which either $P F_{1}-P F_{2}=6$ or $P F_{2}-P F_{1}=6 ;$ point $P(3,0)$ is one such point, Then sketch the hyperbola.(FIGURE CANNOT COPY)
Use the Distance Formula to show that the equation of the parabola with focus $F(0,1)$ and directrix $y=-1$ is $$y=\frac{1}{4} x^{2}$$
Use the Distance Formula to show that the equation of the parabola with focus $F(0,2)$ and directrix $y=-2$ is $y=\frac{1}{8} x^{2}$
Following a $90^{\circ}$ counterclockwise rotation about the origin, the image of $A(3,1)$ is point $B(-1,3) .$ What is the image of point $A$ following a counterclockwise rotation ofa) $180^{\circ}$ about the origin?b) $270^{\circ}$ about the origin?c) $360^{\circ}$ about the origin?
Consider the point $C(a, b) .$ What is the image of $C$ after a counterclockwise rotation ofa) $90^{\circ}$ about the origin?b) $180^{\circ}$ about the origin?c) $360^{\circ}$ about the origin?
Given the point $D(3,2),$ find the image of $D$ after a counterclockwise rotation ofa) $90^{\circ}$ about the point $E(3,4)$b) $180^{\circ}$ about the point $F(4,5)$c) $360^{\circ}$ about the point $G(a, b)$