Coordinate Geometry Packet
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Math 512
Coordinate Geometry
Math 512
3
P
2
(x2,y2)
M
(x1,y1)
1
Q
-4
-2
2
4
-1
-2
-3
Given: Points P (x1,y1) and Q(x2,y2), write the following
Formulas
Midpoint:
M=
Slope of PQ =
Distance from P to Q =
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Coordinate Geometry Packet
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Lines in Triangles
Math 512
Math 512
Write the definitions of:
Altitude
Median
Perpendicular Bisector
In ΔABC, draw in the altitude from C (to AB)
Draw in the median from B (to AC)
Draw in the perpendicular bisector of AB
B
A
C
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Coordinate Geometry Packet
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PRACTICE WITH EQUATIONS OF LINES
Slope-intercept form: y = mx + b
Point-slope form:
y " y1 = m( x " x1 )
!
For each of the following
!
m=slope b=y intercept
( x1, y1) = point on the line
a) Graph the line described
b) Write an equation of the line
!
1. slope = -2
point =(0, -4)
2. slope = 2/3
point = (-6, -2)
3. slope = ½
point = (-4, 5)
4. point1= (1, 5)
point2 = (5, 13)
5. point1= (-4, -1)
point2 = (-1, 11)
6. point1= (0, 3)
point2 = (2, 9)
7. point1= (-2, 5)
point2 = (-2, 0)
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Math 512
Coordinate Geometry 14
Packet
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Math 512
12
10
8
6
4
2
-10
-5
5
10
15
-2
14
12
10
8
6
4
2
-10
-5
5
10
15
-2
14
12
10
8
6
4
2
-10
-5
5
10
-2
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15
Coordinate Geometry Packet
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Math 512
6
6
4
4
2
2
-5
5
10
-5
5
-2
-2
-4
-4
6
6
4
4
2
2
-5
5
10
-5
10
5
-2
-2
-4
-4
10
6
6
4
4
2
2
-5
5
10
-5
5
-2
-2
-4
-4
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10
Coordinate Geometry Packet
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Math 512
ALTITUDES, MEDIANS AND PERPENDICULAR BISECTORS
1. a) On the grid, draw the line segment joining S ("3,3) and T ( 7,7) .
b) Draw the perpendicular bisector of segment ST.
10
!
!
c) Find the equation of the perpendicular
bisector.
8
6
4
2
-10
-5
5
10
-2
-4
2. Graph ΔABC using vertices A("3,1), B(5,"1) and (9,9) .
a) Draw the median from A to the midpoint of BC.
10
b) Draw the altitude from C to AB.
!
8
c) Draw the perpendicular bisector of AC.
d) Find the equations for the
three lines you drew in a, b and c.
6
4
2
-10
-5
5
-2
-4
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10
Coordinate Geometry Packet
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Math 512
3. P ("6,"4 ), Q(6,2) and R("2,8) are the vertices of a triangle.
a) Find the equations of the three sides.
!
8
6
b) Find the equation of the perpendicular
bisector of each side.
4
2
-10
-5
5
-2
-4
c) Find the point of intersection, A, of the
perpendicular bisectors, and the lengths of the
segments AP, AQ, and AR
8
4. P ("1,3), Q(0,6), R( 3,5), S ( 3,0) are the vertices
of a quadrilateral.
a) Find the equations of the diagonals.
6
!
4
b) Is PR the perpendicular bisector of QS?
2
c) Is QS the perpendicular bisector of PR?
-10
-5
5
-2
d) Is PQRS a rhombus? A parallelogram?
-4
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Coordinate Geometry Packet
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Math 512
FINDING EQUATIONS OF LINES IN TRIANGLES
NOTES
ALTITUDE
MEDIAN
PERPENDICULAR BISECTOR
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Coordinate Geometry Packet
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LINES IN TRIANGLES
1. Triangle ABC has A = ("4,"2), B = (0,2), C = ("8,6) .
Plot ABC.
a) Find M, the midpoint of BC, and
N, the midpoint of AC.
!
b) Write an equation for AM; for BN.
c) Find the point of intersection of AM and BN.
d) Find the lengths of AM and BN.
e) Find the slopes of MN and AB.
f) Write an equation for the altitude from A.
2. Triangle ABC has A = ("1,3), B = ( 4,6), C = ( 3,"3)
Plot ABC.
a) Find M, the midpoint of AB and
N, the midpoint of BC.
!
b) Find the lengths of MN and AC.
c) Show that the three medians all pass
through the point Q=(2, 2).
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Math 512
Coordinate Geometry Packet
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Math 512
MORE PRACTICE WITH LINES IN TRIANGLES
In ΔABC, point A= (-2, 1) point B= (8, 11) and point C=(3, -4)
1. Plot the three vertices and draw the line segments for ΔABC.
2. Find the length of each side of ΔABC:
AB=
, BC =
AC=
3. Write the equation for sides AB and AC, each in y=mx+b (slope-intercept) form:
AB
AC
4. Write the equation for BC in point-slope form, two different ways:
BC
or
5. Draw the altitude from A to BC and write its equation in point-slope form:
Altitude:
6. Find the point where the altitude from A intersects BC and label it D =(
,
)
Check that it satisfies the equations for both the altitude and for BC;
7. Find the length of AD =
8. Write an equation of the median from C to AB
Median:
9. Prove, two different ways that ΔABC is a right triangle.
10. Find the area of ΔABC two different ways. Make sure that you get the same answer both
ways.
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Coordinate Geometry Packet
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Math 512
Coordinate Geometry Packet
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TYPES OF TRIANGLES
1. Given that points A = (0,0), B = ( 4,8), C = (6,2)
are the vertices of ΔABC. Plot ABC.
a) Show that ΔABC is an
isosceles triangle.
!
b) Find the coordinates of D,
the midpoint of the base.
c) Show that CD is perpendicular to AB.
2. Given ΔRST with vertices
R = (0,6), S = (2,0), T = (8,2) . Plot ΔRST
!
a) Show that ΔRST is a right triangle
and state a reason for your conclusion.
b) If SM is a median to RT, show that
SM=RM=TM.
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Math 512
Coordinate Geometry Packet
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3. Given ΔPQR with vertices
P = (2,4 ), Q = (5,8), R = (9,5)
Plot ΔPQR
!
a) Show that ΔPQR is isosceles.
b) Show that ΔPQR is a right triangle.
4. The vertices of ΔLMN are
L = ( 3,"1), M = ( 7,3), N = ("1,7) .
Plot ΔLMN.
!
a) Show that ΔLMN is isosceles.
b) Show that ΔLMN is not equilateral.
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Math 512
Coordinate Geometry Packet
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Math 512
USING COORDINATE GEOMETRY TO PROVE A SHAPE
IS A PARTICULAR KIND OF POLYGON
NOTES
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Coordinate Geometry Packet
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Math 512
TYPES OF QUADRILATERALS
!
1. Plot the points
A = (1,1), B = (5,3), C = ( 7,5), D = ( 3,3) .
Use the distance formula to show that the
opposite sides of quadrilateral ABCD are
congruent. What kind of quadrilateral is ABCD?
2. Plot the points
A = ("7,2), B = ("2,5), C = (1,2), D = ("4,"1) .
Use the slope formula to show that ABCD is a
parallelogram.
!
!
3. Plot the points
A = ("5,"2), B = ("3,4 ), C = (5,6), D = ( 3,0) .
Use the midpoint formula to show that the
diagonals AC and BD have the same midpoint.
What kind of quadrilateral is ABCD?
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Coordinate Geometry Packet
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4. Plot the points A = (2,3), B = ( 3,"4 ), C = (8,1), D = ( 7,8) .
Show that segments AC and BD perpendicular bisect each other.
What kind of quadrilateral is ABCD?
!
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Math 512
Coordinate Geometry Packet
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Math 512
Finding Points at a Given Distance from another Point
Example 1:
8
6
4
2
-10
-5
5
10
-2
-4
-6
-8
We can solve this problem algebraically or geometrically. Let’s look at the algebraic solution
first.
We’ll look at the graph to do the geometric solution.
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Coordinate Geometry Packet
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Math 512
EXAMPLE 2:
12
Solve this one using whichever method you liked better.
10
8
A
6
4
2
-5
5
Problem #3: Point Q is on the y-axis and is 2 5 units from (4, 1). Find the coordinates of Q.
4
!
2
A
-5
5
-2
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-4
Coordinate Geometry Packet
Problem #4:
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Math 512
Point A (4, 5) is 10 units from point B (-4, k). Solve for k.
10
8
6
A
4
2
-5
5
10
-2
Problem #5: Point M (3,3) is 15 units from point N (k, 12). Solve for k.
18
16
14
12
10
8
6
4
2
-10
-5
A
5
-2
-4
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10
15
20
Coordinate Geometry Packet
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Math 512
Distance from a Point to a Line
CLASSWORK
1. Graph the point P= (1, 5) and the line x + y = "2
Find the points on the line which have x- coordinates -5, –4, -3, -2, and -1,
call them A, B, C, D, and E respectively
Find as decimals to the nearest hundredth the lengths.
!
AP=
BP=
6
CP=
4
DP=
EP=
2
How do we define the distance between the point P
and the line y = "x " 2 ?
-5
5
-2
!
-4
-6
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Coordinate Geometry Packet
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Math 512
1
x + 2 . Find the equation of the line perpendicular to this line that goes
2
through the point A= (6,0). Graph this line on the same set of axes and determine where the
two lines meet, call this point B. Find the length of AB. This is the distance from the point A to
the original line.
!
2. Graph the line y =
6
4
2
-5
5
-2
3
3. Graph the line y = " x + 5 . Find the equation of the line perpendicular to this line that goes
2
through the point D= (-4,-2). Graph this line on the same set of axes and determine where the
two line meet, call this point E. Find the length of DE. This is the distance from the point D to
the original line.
6
!
4
2
-5
5
-2
-4
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Coordinate Geometry Packet
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Math 512
HOMEWORK
4. Find the distance from (-5, 4) to y =
1
x "1.
3
6
4
!
2
-5
5
-2
-4
5. Find the distance from (5, 7) to 3x " 4 y = 12 .
6
!
4
2
5
-2
-4
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10
Coordinate Geometry Packet
6.
a)
b)
c)
d)
e)
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Math 512
The triangle ARE has vertices at A=(0,1), R= (8, -1) and E= (10, 7).
Find the equation of the line AE.
Find the equation of the altitude from R to AE.
Find the length of the altitude from R to AE.
Find the area of triangle ARE.
Find the equation of the median from A to ER.
10
8
6
4
2
5
-2
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10
15
Coordinate Geometry Packet
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Math 512
USING COORDINATE GEOMETRY TO DO PROOFS
ABOUT GENERAL SHAPES
How do we plot a general shape using coordinate geometry? Rather than
assigning numbers to the coordinates, we use variables. The relationships
among the variables determine the properties of the shape.
EXAMPLE
Use the distance formula to decide which of these triangles has two sides of equal length.
Use slopes to determine which of the following quadrilaterals are parallelograms.
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Coordinate Geometry Packet
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Math 512
How could we use coordinates to represent a rectangle?
How do we do a proof?
Example:
Prove that the line segments joining the midpoints of the successive sides of any
quadrilateral form a parallelogram.
First: Make a diagram and label the coordinates:
Second: Write the Given: and To Prove: in terms of your diagram
Use our formulas to prove what is required.
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Coordinate Geometry Packet
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Math 512
Practice with Coordinate Geometry Proofs
1. Show that the diagram to the right
represents an isosceles trapezoid:
You must show that two sides are
parallel and the other two sides have the
same length.
(a-b, c)
2. Here is a theorem:
The length of the diagonals of an isosceles trapezoid are equal.
a) For this theorem, write (in terms of the labels in the diagram above), what is
Given:
To Prove:
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Coordinate Geometry Packet
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Math 512
Proofs for you to do:
1) Prove that the line segment joining the midpoints of two sides of a triangle is
parallel to the third side and has length half that of the third side.
2) Prove that the medians to the legs of an isosceles triangle are equal in length.
3) Prove that the length of the diagonals of a rectangle are equal.
4) Prove that the line segments joining the midpoints of the successive sides of
any rectangle form a rhombus.
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