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Section 1
The Coordinate Plane
Plot the given points in a coordinate plane:$$(2,3),(-2,3),(4,5),(4,-5),(-4,5),(-4,-5)$$
Find the coordinates of the points shown in the figure.(GRAPH NOT COPY)
$3-6$ A pair of points is graphed.(a) Find the distance between them.(b) Find the midpoint of the segment that joins them.(GRAPH NOT COPY)
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(0,8),(6,16)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(-2,5),(10,0)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(-3,-6),(4,18)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(-1,-1),(9,9)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(6,-2),(-1,3)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(-1,6),(-1,-3)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(3,4),(-3,-4)$$
$7-14$ A pair of points is graphed.(a) Plot the points in a coordinate plane.(b) Find the distance between them.(c) Find the midpoint of the segment that joins them.$$(5,0),(0,6)$$
Draw the rectangle with vertices $A(1,3), B(5,3), C(1,-3),$ and $D(5,-3)$ on a coordinate plane. Find the area of the rectangle.
Draw the parallelogram with vertices $A(1,2), B(5,2)$ $C(3,6),$ and $D(7,6)$ on a coordinate plane. Find the area of the parallelogram.
Plot the points $A(1,0), B(5,0), C(4,3),$ and $D(2,3),$ on a coordinate plane. Draw the segments $A B, B C, C D,$ and $D A$ . What kind of quadrilateral is $A B C D,$ and what is its area?
Plot the points $P(5,1), Q(0,6),$ and $R(-5,1),$ on a coordinate plane. Where must the point $S$ be locatedso that the quadrilateral $P Q R S$ is a square? Find the area of this square.
$19-32$ Sketch the region given by the set.$$\{(x, y) | x \leq 0\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | y \geq 0\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | x=3\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | y=-2\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | 1 < x < 2\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | 0 \leq y \leq 4\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | x y<0\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | x y>0\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) | x \geq 1 \text { and } y<3\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y) |-2 < x < 2 \text { and } y \geq 3\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y)| | x |>4\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y)| | y | \leq 2\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y)| | x | \leq 2 \text { and }|y| \leq 3\}$$
$19-32$ Sketch the region given by the set.$$\{(x, y)| | x |>2 \text { and }|y|>3\}$$
Which of the points $A(6,7)$ or $B(-5,8)$ is closer to the origin?
Which of the points $C(-6,3)$ or $D(3,0)$ is closer to the point $E(-2,1) ?$
Which of the points $P(3,1)$ or $Q(-1,3)$ is closer to the point $R(-1,-1) ?$
(a) Show that the points $(7,3)$ and $(3,7)$ are the same distance from the origin.(b) Show that the points $(a, b)$ and $(b, a)$ are the same distance from the origin.
Show that the triangle with vertices $A(0,2), B(-3,-1)$ and $C(-4,3)$ is isosceles.
Find the area of the triangle shown in the figure.(GRAPH NOT COPY)
Refer to triangle $A B C$ in the figure.(a) Show that triangle $A B C$ is a right triangle by using the converse of the Pythagorean Theorem (see page 292$) .$(b) Find the area of triangle $A B C .$
Show that the triangle with vertices $A(6,-7), B(11,-3)$ , and $C(2,-2)$ is a right triangle by using the converse of the Pythagorean Theorem. Find the area of the triangle.
Show that the points $A(-2,9), B(4,6), C(1,0),$ and $D(-5,3)$ are the vertices of a square.
Show that the points $A(-1,3), B(3,11),$ and $C(5,15)$ are collinear by showing that $d(A, B)+d(B, C)=d(A, C)$
Find a point on the $y$ -axis that is equidistant from the points $(5,-5)$ and $(1,1) .$
Find the lengths of the medians of the triangle with vertices $A(1,0), B(3,6),$ and $C(8,2) .$ (A median is a line segment from a vertex to the midpoint of the opposite side.)
Find the point that is one-fourth of the distance from the point $P(-1,3)$ to the point $Q(7,5)$ along the segment $P Q$ .
Plot the points $P(-2,1)$ and $Q(12,-1) .$ Which (if either) of the points $A(5,-7)$ and $B(6,7)$ lies on the perpendicular bisector of the segment $P Q ?$
Plot the points $P(-1,-4), Q(1,1),$ and $R(4,2),$ on a coordinate plane. Where should the point $S$ be located so that the figure $P Q R S$ is a parallelogram?
If $M(6,8)$ is the midpoint of the line segment $A B,$ and if $A$ has coordinates $(2,3),$ find the coordinates of $B .$
(a) Sketch the parallelogram with vertices $A(-2,-1)$ $B(4,2), C(7,7),$ and $D(1,4) .$(b) Find the midpoints of the diagonals of this parallelogram.(c) From part (b) show that the diagonals bisect each other.
The point $M$ in the figure is the midpoint of the line segment $A B S$ how that $M$ is equidistant from the vertices of triangle $A B C .$(GRAPH NOT COPY)
Distances in a City $\quad$ A city has streets that run north and south, and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points $A$ and $B$ is 7 blocks $-$ that is, 3 blocks east and 4 blocks north. To find the straight-line distances $d,$ we must use the Distance Formula.(a) Find the straight-line distance (in blocks) between $A$ and $B$ .(b) Find the walking distance and the straight-line distance between the corner of 4 th St. and 2 nd Ave. and the corner of 11 th St. and 26 th Ave.(c) What must be true about the points $P$ and $Q$ if the walking distance between $P$ and $Q$ equals the straight-line distance between $P$ and $Q ?$
Halfway Point Two friends live in the city described in Exercise $51,$ one at the corner of 3 rd St. and 7 th Ave, the other at the corner of 27 th $\mathrm{St}$ . and 17 th Ave. They frequently meet at a coffee shop halfway between their homes.(a) At what intersection is the coffee shop located?(b) How far must each of them walk to get to the coffee shop?
Pressure and Depth The graph shows the pressure experienced by an ocean diver at two different depths. Find and interpret the midpoint of the line segment shown in the graph.(GRAPH NOT COPY)
Shifting the Coordinate Plane Suppose that each point in the coordinate plane is shifted 3 units to the rightand 2 units upward.(a) The point $(5,3,3)$ is shifted to what new point?(b) The point $(a, b)$ is shifted to what new point?(c) What point is shifted to $(3,4) ?$(d) Triangle $A B C$ in the figure has been shifted to triangle $A^{\prime} B^{\prime} C^{\prime} .$ Find the coordinates of the points $A^{\prime}, B^{\prime},$ and $C^{\prime} .$(GRAPH NOT COPY)
Reflecting in the Coordinate Plane Suppose that the $y$ -axis acts as a mirror that reflects each point to the right of it into a point to the left of it.(a) The point $(3,7)$ is reflected to what point?(b) The point $(a, b)$ is reflected to what point?(c) The point is reflected to what point?(d) Triangle $A B C$ in the figure is reflected to triangle $A^{\prime} B^{\prime} C^{\prime} .$ Find the coordinates of the points $A^{\prime}, B^{\prime},$ and $C^{\prime} .$
Completing a Line Segment Plot the points $M(6,8)$ and $A(2,3)$ on a coordinate plane. If $M$ is the midpoint of the line segment $A B,$ find the coordinates of $B$ . Write a brief description of the steps you took to find $B,$ and your reasons for taking them.
Completing a Parallelogram Plot the points $P(0,3)$ $Q(2,2),$ and $R(5,3)$ on a coordinate plane. Where should the point $S$ be located so that the figure $P Q R S$ is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.