Determine whether or not (-4,2) lies on the graph of each equation.
1. 4 x  2 y  20
2.
x y
 0
4 2
b g
3. y  2  
1
x4
3
4. y  2  
1
x4
3
5.
6.
b g
bx  4g by  2g 16
bx  4g by  2g 16
2
7. y  4 x  2
8. y  x  2
9. y  x  2
10. x  y  2
2
Determine whether or not (-4,2) lies in the shaded region determined by the inequality
1.
2.
3.
bx  4g by  2g 16
bx  4g by  2g 16
bx  4g by  2g 16
2
2
2
2
2
2
4. y  4 x  2
5.
y  x 3
6. x 2  y 2  25
7. x 2  y 2  16
Write an equation of the line that is perpendicular to the line with
equation 4 x  3y  10 and which passes through the midpoint of the
segment with endpoints 8,2 & 4,6
b gb g
ABC has vertices
the point that is
2
of
3
b gb g b g. Find the coordinates of
A 0,0 , B 4,8 & C 8,14
the way from A to the midpoint of
BC
AB has
slope

4
.
3
The coordinates of A are
b2,4gand the
coordinates of B are b
, find the value of k.
13, k g
ABC has vertices
b gb g b g.
A 2,5 , B 8,7 & C 2,1
Compute each of the following:
The slope of
The slope of
The slope of


AB 


AC 

CB
The slope of the line connecting the midpoints of AB& BC 
The slope of the line connecting the midpoints of AB& AC 
The slope of the line connecting the midpoints of AC& BC 
The slope of the altitude from B to
AC 
The slope of the altitude from A to
BC 
The slope of the altitude from C to
AB 
Explain what is going on.
AS quickly as possible, name the coordinates of exactly one
point on each line.
a.
y  5x  3
b.
y 1  5 x  3
c.
y 8  
d.
2 x  13y  17
e.
y
b g
b g
1
x6
2
F
I
G
H J
K
17
12
1

x
50
19
10
Estimated time 15 seconds. If it took more than 30 seconds, you are missing something critical.
(
,
)
(
,
)
(
,
)
(
,
)
(
,
)
Find an equation of the line with the same x-intercept as
5
3
2 x  y  10 and the same y-intercept as x  y  4  0
3
7
Find an equation for the altitude from A in ABC with vertices
having the following coordinates: Ab
2,6gb
, B 10,8g
& Cb
8,0g
Is this altitude also a median. Explain how you know.
Find the coordinates of the point of intersection of the lines with
equations 3x  5y  17 and 7 x  4 y  70
A few coordinate geometry problems to try:
1. Find the midpoint of the segment with endpoints
& 23,4g
b7,18gb
2. The midpoint of a segment has coordinates b
2,6gOne
of the endpoints of the segment has coordinates
b4,3g. Find the coordinates of the other endpoint.
3.
b g b gare the endpoints of a segment. Find
A 5,11 & B 7,13
the coordinates of point F on
AB
so that AF:FB = 5:7
4. Write an equation of the line through
& 4,13g
b7,11gb
5. Write an equation of the line through b
7,13gthat is
parallel to the line with equation 5x  3y  17
7,4gthat is
6. Write an equation of the line through b
perpendicular to the line with equation 2 x  y  7
7. Write an equation of the perpendicular bisector of
8,5gb
& 12,1g
the segment with endpoints b
8. Find the distance from
b3,11gto b9,6g
9. Given the following points and their coordinates:
Ab
5,19gb
, B 3,7g
& Cb
7,9gfind the length of the median to BC
in ABC
Match each line with the corresponding equation and table
y
1
2
I. y  x  2
B
C
II. y  3x  1
III y  4  
x
b g
1
x2
2
b g
IV y  7  3 x  1
D
A
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
LINE 1
y
-14
-11
-8
-5
-2
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
GRAPHS
A
B
C
D
LINE 2
y
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
LINE 3
y
19
16
13
10
7
4
1
-2
-5
-8
-11
-14
-17
-20
-23
-26
-29
-32
-35
-38
LINE 4
y
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-9
-9.5
-10
EQUATIONS
TABLES



AB and AC are tangent to the circles. AB is a vertical line. Find
the coordinates of A
A
C(5,9)
B
O(2,5)
a) Draw the path of a golf shot (
Miniature golf) which would put the
ball in the hole.
Hole
b) Now assume that there is a
barrier between A and B. then draw
the path of a successful shot.
A
.
.
B
Ball
Suppose that the midpoint of the segment on the floor which
passes exactly across the doorway is the origin of a 3dimensionsl coordinate system. East is the x-direction (towards
the Lake), North is y ( towards Church street), and "up" is z.
The coordinates are listed (x,y,z) Estimate the co-ordinates of
the middle of the lens on the overhead projector, in feet.
Estimate the co-ordinates of the entrance to the main office, in
feet. Estimate the coordinates of the point in Lake Michigan
closest to us, in feet.
Where would the following points be? (-2,10,5) and (-2,4,-10)?
A circle is tangent to the line with equation y  2 x and to the
line with equation y  2 x  10 . Its center lies on the line with
equation y  3x  20 . Find an equation of the circle.
List the co-ordinates of 12 points on the graph of
bx  21g by  19g 17
2
2
2
A( 2,5), B(6,1) Chord AB is 5 units from the center of the circle;

Find the coordinates of the centers of both circles

Find the area of both circles

Find the length of the common tangent of the circles

Find the coordinates of the points where the tangents
intersect the circles
R
||y  2
|y  3 x  8 ?
How many circles are tangent to all 3 of the lines S
|| 4 3
|Ty   4 x  8
Find the center and radius of one of these circles.
b gb gb g
Find the area of the triangle with vertices 2,5 , 6,12 & 8,4
b gb g
Find the area of the circle with equation x  37 2  y  13 2  17
b6,17gis a point on the circle with center b1,5g. Write an
equation of the circle.
Write an equation of the line tangent to the circle above
passing through 6,17
b g
Find the center and radius of the circle with equation
5x 2  15x  5y 2  3y  6
& 1,2gare the coordinates of points that are opposite
b5,8gb
vertices of a square. Find the coordinates of the other two
vertices of the square.
Find the length of the longest altitude of ABC given
coordinates A(2,5), B 6,8 & C 6,13
bg b g
Find the area of the region enclosed by the x-axis, and the
y  2x
R
|| 1
y  x 5
graphs of the following equations : S
||4 x 3y  44
T
Find the area of the region enclosed by the solution to the
system
R
|Sx  y  16
|Tx  4 x  y  0
2
2
2
2
b gb gb g
Find the area of the  with vertices 2,8 , 6,4 & 0,4 . Find the
length of the shortest altitude of this triangle.
1
R
y2  b
x  4g
|
3
Find the intersection of the lines S
||y  5  1 bx  3g
T 2
Find the height of the trapezoid with vertices at
& 12,6g
b0,0g, b0,8g, b4,10gb
Find the point where the lines with equations
2 x  5y  10 & 4 x  13  10 y cross. Then find the point where the lines
with equations 2 x  5y  10 & 4 x  13  10 y cross
bg bg
A(0,0), B 4,6 ,& C 8,2 are the vertices of a triangle. Find the
coordinates of the point where the altitude from B intersects
the altitude from A?
Find an equation of the line joining the intersection of
x  y  19
R
S
Tx  y  7
and
y  6x  5
R
S
Tx  8 y  7
Rx  y  100
Find all points of intersection of S
T4 y  3x
2
2
Find the distance between the lines with equations
y  3x  5 and y  3x  5
Find an equation of the circle that passes through the points
with coordinates 2,7 ; 8,7 & 9,0
b gb gb g
Where does the line with slope
b g
1
that passes through 2,6
2
intersect the line with slope 3 and y-intercept -13?
Where does the median from A intersect the median from B?
Where does the median from A intersect the median from C?
y
A(0,6)
B(10,0)
x
C(-4,-6)
Find the coordinates of the point of reflection of the point (0,8)
over the line with equation x  2 y  4
Write a mathematical description of the region shaded on the
left of the line. Be as specific as possible.
y
(0,4)
(2,0)
x
 2,5 ,  8, 2 , 3,1 and  6, 8
are the coordinates of the vertices of a quadrilateral.
Determine the area of the quadrilateral.
 0,6 , 12,0 and the origin are vertices of a triangle.
Determine the area of the triangle.
Determine the area of the triangle formed by connecting the midpoints of the sides of the
triangle. What observation can you make about the area of the two triangles?
1
Determine the coordinates of the point of the way from  0, 0  to  0, 6 
3
1
Determine the coordinates of the point of the way from  0, 6  to 12,0
3
1
of the way from 12,0 to  0, 0 
3
Determine the area of the triangle whose vertices are the three points you just determined.
Any further observations or conjectures?
Determine the coordinates of the point
1
1
x  8 and y= x  4 . The
2
2
two circles are also tangent to each other. Determine the length of one of the common
external tangent segments.
Two circles are both tangent to the lines with equations y 
 8, 6 ,  3,6 , 5,12 and  0,0
are the vertices of a quadrilateral. Determine the
following .
a) The area of the quadrilateral
b) The intersection point of the diagonals of the quadrilateral
c) The midpoint of the longest diagonal of the quadrilateral
d) An equation of a circle that contains all of the vertices of the quadrilateral
e) An equation of the smallest circle that contains at least one point form each side os the
quadrilateral.