Download Geometry Notes Ch 13

Chapter Thirteen
Coordinate Geometry
USE GRAPH PAPER FOR HOMEWORK IN
THIS CHAPTER, AS NECESSARY!
Make sure you properly annotate all
coordinate graphs in this chapter!
Objectives
A. Use the terms defined in the chapter
correctly.
B. Properly use and interpret the symbols
for the terms and concepts in this chapter.
C. Appropriately apply the postulates and
theorems in this chapter.
D. Understand and apply the distance and
midpoint formulas.
E.
F.
G.
H.
I.
Calculate and use the slopes of lines.
Perform the basics of vector mathematics.
Graph linear equations
Write the equations of straight line graphs
Organize and write a coordinate proof.
Section 13-1
The Distance Formula
Homework Pages 526-527:
1-24, 28, 36
Exclude 14, 16
Objectives
A. Understand and apply the terms
‘origin’, ‘axes’, ‘quadrants’, and
‘coordinate plane’.
B. Properly graph points, lines, and
circles on a coordinate plane.
C. Derive and utilize the distance
formula.
D. Understand the components of the equation of a circle.
E. Graph a circle in a coordinate plane based on its equation.
The Coordinate Plane
• coordinate plane: plane formed by the intersection of a
horizontal and a vertical real number line, called the
coordinate axes, where every point in the plane can be
represented by an ordered pair of real numbers, called its
coordinates.
• x-axis: the horizontal coordinate axis
• y-axis: the vertical coordinate axis
• origin: intersection of the coordinate axes
Coordinate Plane
y-axis
5
3
2
1
-2 -1
-2
(8, 5)
origin
1 2 3
8
x-axis
The Coordinate Plane - Quadrants
Quadrants: four regions of the coordinate plane created by the two axes.
y
Quadrant II:
x is negative; y is positive
Quadrant I:
Both x & y are positive
x
Quadrant III:
Both x & y are negative
Quadrant IV:
x is positive; y is negative
Building the Distance Formula
1. Label the points.
P1=(x1,y1)
P2=(x2,y2)
2. Label the coordinates
P1=(3,2)
P2=(6,3)
3. Substitute values
3
2
1
-2 -1
-2
2 )
(xP2,y
2
(xP11,y1)
1 2 3
|y2 – y1|
6
Distance between points
on x-axis, |x2 – x1|
Building the Distance Formula
c2  a 2  b2
(6,3)
c  a 2  b2
c
|y2 – y1|
x2  x1 2   y2  y1 2
(3,2)
|x2 – x1|
Theorem 13-1 (Distance Formula)
The distance d between points (x1, y1 ) and (x2, y2 ) is given
by: d 
2
2
x

x

y

y
 2 1  2 1
(x2 , y2)
d
(x1 , y1)
x2 - x1
d 2   x 2  x 1 2   y 2  y1 2
y2 - y1
 Theorem 13-2 (Standard Form for the equation of a Circle)
An equation of a circle with center (a, b) and radius r
is (x - a)2 + (y - b)2 = r2.
(x, y)
r
(a, b)
x-a
y-b
Sample Problems
Find the distance between the points. If necessary, you may draw
graphs but you should NOT need to use the distance formula.
1. (- 2, - 3) & (- 2, 4)
What should you do first? Plot the points.
y
What is the distance between these 2 points?
7 units
(-2, 4)
3
2
1
1 2 3
(-2, -3)
x
Sample Problems
Find the distance between the points. If necessary, you may draw
graphs but you should NOT need to use the distance formula.
3. (3, - 4) & (- 1, - 4)
What should you do first? Plot the points.
y
What is the distance between these 2 points?
4 units
3
2
1
x
1 2 3
(-1, -4)
(3, -4)
Sample Problems
Use the distance formula to find the distance between the 2 points.
What should you do first?
5. (- 6, - 2) & (- 7, - 5)
P1 = ? P 2 = ?
P1 = (- 6, - 2) P2 = (-7 , - 5)
x1 = ? y1 = ? x2 = ? y2 = ?
x1 = - 6, y1 = - 2, x2 = - 7, y2 = - 5
What is the distance formula?
d   x 2  x1  2   y 2  y1  2
y
3
2
1
x
1 2 3
(-6, -2)
(-7, -5)
d
 7   6    5   2 
d
  1    3 
2
2
2
2
d  1  9  1 0 u n its
Sample Problems
Use the distance formula to find the distance between the 2 points.
What should you do first?
7. (- 8, 6) & (0, 0)
P1 = ? P 2 = ?
P1 = (0, 0) P2 = (- 8 , 6)
x1 = ? y1 = ? x2 = ? y2 = ?
x1 = 0, y1 = 0, x2 = - 8, y2 = 6
What is the distance formula?
d   x 2  x1  2   y 2  y1  2
y
(-8, 6)
10
3
2
1
6
8
(0, 0)
1 2 3
x
d
 8  0    6  0 
2
2
d  64  36
d  1 0 0  1 0 u n its
A triangle with sides 6, 8, and 10. Sound familiar?
Sample Problems
Find the distance between the points named. Use any method
you choose.
9. (5, 4) & (1, - 2)
11. (- 2, 3) & (3, - 2)
Sample Problems
Given the points A, B & C. Find AB, BC & AC. Are A, B &
C collinear? If so, which point is in the middle?
13. A(0, 3) B(- 2, 1) C(3, 6)
15. A(- 5, 6) B(0, 2) C(3, 0)
Sample Problems
Find the center and the radius of each circle.
17. (x + 3)2 + y2 = 49
What is the Standard Form for the equation of a circle?
An equation of a circle with center (a, b) and radius r is
(x - a)2 + (y - b)2 = r2.
(x + 3)2 + y2 = 49
(x + 3)2 + (y – 0)2 = 49
x–a=x+3 a=-3
y – b = y - 0 b = 0 center (a, b) = center (-3, 0)
r2 = 49
radius = 7 units
Sample Problems
Find the center and the radius of each circle.
19. (x - j)2 + (y + 14)2 = 17
What is the Standard Form for the equation of a circle?
An equation of a circle with center (a, b) and radius r is
(x - a)2 + (y - b)2 = r2.
(x - j)2 + (y + 14)2 = 17
x–a=x-j
y – b = y + 14
r2 = 17
a=j
b = -14
center (a, b) = center (j, -14)
radius = 17 units
Sample Problems
Write an equation of the circle with the given center and radius.
21. C(3, 0) r = 8
23. C(- 4, - 7) r = 5
What is the Standard Form for the equation of a circle?
An equation of a circle with center (a, b) and radius r is
(x - a)2 + (y - b)2 = r2.
center (3, 0) = center (a, b)
a = ? b = ? a = 3, b = 0
(x - 3)2 + (y - 0)2 = 82
(x - 3)2 + y2 = 64
center (-4, -7) = center (a, b)
a = ? b = ? a = -4, b = -7
(x – (-4))2 + (y – (-7))2 = 52
(x + 4)2 + (y + 7)2 = 25
Section 13-2
Slope of a Line
Homework Pages 532-533:
1-24
Excluding 10
Objectives
A. Understand the terms ‘linear
equation’ and ‘slope of a line’.
B. Understand and identify lines with
positive, negative, zero, and
undefined slopes.
C. Calculate the slopes of various
lines.
D. Use the slope of a line to graph linear equations.
 Linear Equation
• A linear equation is any equation where the graph of the
solution set is a line.
• Example: y = 2x (How many solutions are there to this
equation?)
Would it be possible to list ALL of the solutions?
SOME solutions!
If x =
0
Then y =
(2x)
0
1
2
2
4
Plot the points.
Draw the line.
y
How many points
are on a line?
2
1
1 2 3
x
 Linear Equation
• NOTE! All values of x and y that satisfy the equation y = 2x form a
point (x, y) that is ON the line.
• NOTE! All coordinates (x, y) of points on the line make the equation
y = 2x true!
• Therefore, the GRAPH represents ALL of the solutions to the
equation!
SOME solutions!
If x =
0
1
Then y =
(2x)
Plot the points.
y
0
2
2
1
1 2 3
2
4
x
Slopes of Lines
rise y y 2  y1


 slope  m 
run x x 2  x1
 lines with positive slope: rise to the right
 lines with negative slope: rise to the left
• steeper line: has a slope with a greater absolute value.
 the slope of a horizontal line: is zero
 the slope of a vertical line: is undefined
(x2 , y2)
(13, 14)
14
rise = y = y2 - y1
y = 14 - 7
7
(x1 , y1)
run = x = x2 - x1
(8, 7)
x = 13 - 8
rise y y2  y1 14  7 7
slope  m 




run x x2  x1 13  8 5
8
13
rise y y 2  y1
slope  m 


run x x 2  x1
slope = 0
slope is undefined
Sample Problems
Find the slope of the line through the given points.
7. (7, 2) & (2, 7)
What should you do first?
y
P2 (2, 7)
Label the points: P1 = (7, 2) P2 = (2, 7)
Label the coordinates of the points:
x1 = 7, y1 = 2, x2 = 2, y2 = 7
Write the formula!
rise y y2  y1
slope  m 


run x x2  x1
3
2
1
P1 (7, 2)
1 2 3
x
Fill in the blanks!
y2  y1 7  2 5
1
slope 


   1
x2  x1 2  7 5
1
Sample Problems
Find the SLOPE and the length of AB
What should you do first?
15. A(0, - 9) & B(8, - 3)
Label the points: P1 = A = (0, -9) P2 = B = (8, -3)
y
-1
-2
-3
Label the coordinates of the points:
x1 = 0, y1 = -9, x2 = 8, y2 = -3
x
1 2 3
P2 (8, -3)
Write the formula!
rise y y2  y1
slope  m 


run x x2  x1
Fill in the blanks!
P1 (0, -9)
y2  y1 3  9 6 3
slope 

 
x2  x1
80
8 4
Sample Problems
Find the slope and the LENGTH of AB
15. A(0, - 9) & B(8, - 3)
Label the points: P1 = A = (0, -9) P2 = B = (8, -3)
y
-1
-2
-3
Label the coordinates of the points:
x1 = 0, y1 = -9, x2 = 8, y2 = -3
x
1 2 3
Write the formula!
P2 (8, -3)
distance  d 
 x 2  x1    y2  y1 
2
2
Fill in the blanks!
P1 (0, -9)
distance  d 
8  0   3  9  
2
82  62  64  36  100  10
2
Sample Problems
Find one point to the left and to the right of the given point on
the same line.
2
17. P(- 3, 0) slope =
5
Plot the point.
y
If numerator of slope is
positive, go up y-units.
(if numerator negative,
go down y-units)
3
‘up’ 2
‘right’ 5
If denominator of slope is
2
1
positive, go right x-units.
(if denominator negative,
x
1 2 3
go left x-units)
New point (2, 2)
Sample Problems
Find one point to the left and to the right of the given point on
the same line.
2
2 2
17. P(- 3, 0) slope =

??
5
5 5
y
If numerator of slope is
negative, go down y-units.
‘up’ 2
3
2
1
If denominator of slope is
negative, go left x-units.
‘right’ 5
New point (-8, -2)
1 2 3
x
Sample Problems
Find the slope of the line through the given points.
3. (1, 2) & (3, 4)
5. (1, 2) & (- 2, 5)
9. (6, - 6) & (- 6, - 6)
11. (- 4, - 3) & (- 6, - 6)
Find the slope and the length of AB
13. A(- 3, - 2) & B(7, - 6)
Sample Problems
Find one point to the left and to the right of the given point on
the same line.
2
17. P(- 3, 0) slope =
5
19. P(0, - 5) slope =  1
4
Show point P, Q and R are collinear by showing PQ and QR
have the same slope.
21. P(- 8, 6) Q(- 5, 5) R(4, 2)
Complete.
3
23. A line with slope passes through points (2, 3) & (10, ?)
4
Section 13-3
Parallel and Perpendicular Lines
Homework Pages 537-538:
1-20
Excluding 14
Objectives
A. Understand the relationship
between the geometric and
algebraic definitions of parallel
and perpendicular lines.
B. Determine if 2 lines are parallel,
perpendicular, or neither.
C. Find the slopes of lines parallel or
perpendicular to a given line.
Geometric versus Algebraic ‘speak’
The number 1 lesson from this chapter is to show the tight
linkage between geometry and algebra.
One key to understanding this ‘linkage’ is to understand the
different ‘dialects’ we speak in the mathematical language.
Relate this to the many ‘dialects’ within the United States.
The vast majority of citizens of the United States speak an
English ‘dialect’. In other words, the roots of our language
are the same whether we are from the deep south or from
the industrial northeast. However, the words we use and
the way we pronounce them are different, thus creating the
‘dialect’. We mean the same thing, but say the things
differently.
Geometric versus Algebraic ‘speak’ - continued
•When we speak
‘geometrically’, we use terms
such as:
–Points
–Lines
–Planes
–Angles
–Polygons
–Solids
•When we speak
‘algebraically”, we use terms
such as:
–Numbers
–Variables
–Equations
–Sets
–Inequalities
Both the geometric and algebraic ‘dialects’ are a part
of the language of mathematics.
Parallel Lines
From Chapter 3, the geometric definition of parallel lines is:
Two coplanar lines that do not intersect.
 Theorem 13-3 give the algebraic definition of parallel
lines:
Two non-vertical lines are parallel if and only if their
slopes are equal.
In other words, if m1=m2, then line 1 is parallel to line 2.
Perpendicular Lines
From Chapter 3, the geometric definition of perpendicular
lines is:
Two coplanar lines that intersect at right angles.
 Theorem 13-4 give the algebraic definition of
perpendicular lines:
Two non-vertical lines are perpendicular if and only if the
product of their slopes is (- 1).
In other words, if (m1 x m2 = -1), then line 1 is
perpendicular to line 2.
Sample Problems
1. Find the slope of (a) AB (b) any line parallel to AB (c) any line
perpendicular to AB. A(- 2, 0) & B(4, 4)
y
3
2
1
A (-2, 0)
What should you do first?
Label the points: A = P1 = (-2, 0); B = P2 = (4, 4)
Label the coordinates of the points:
x1 = -2, y1 = 0, x2 = 4, y2 = 4
rise y y2  y1
slope

m



B (4, 4)
run x x2  x1
y y
40 4 2
slope  2 1 
 
x2  x1 4  2 6 3
Slope of a parallel line?
1 2 3
x
Two non-vertical lines are parallel if and
only if their slopes are equal.
slope of parallel line 
2
3
Sample Problems
1. Find the slope of (a) AB (b) any line parallel to AB (c) any line
perpendicular to AB. A(- 2, 0) & B(4, 4)
y
3
2
1
A (-2, 0)
What should you do first?
Label the points: A = P1 = (-2, 0); B = P2 = (4, 4)
Label the coordinates of the points:
x1 = -2, y1 = 0, x2 = 4, y2 = 4
rise y y2  y1
slope

m



B (4, 4)
run x x2  x1
y y
40 4 2
slope  2 1 
 
x2  x1 4  2 6 3
1 2 3
m1  m2  1
x
Slope of a perpendicular line?
Two non-vertical lines are perpendicular if
and only if the product of their slopes is (- 1).
2
 m2  1
3
slope of perpendicular line  
3
2
Sample Problems
E(2, 7)
3. OEFG is a parallelogram.
What is the slope of each side?
What else do you know?
What are the coordinates of point O? O (0, 0)
Label the coordinates of the points:
x1 = 0, y1 = 0, x2 = 2, y2 = 7
rise y y2  y1


run x x2  x1
slope EO  ? slope EO  7  0  7
20 2
O
slope  m 
slope OG  ? slope OG  0
FG || EO so?
slope FG  ? slope FG  7
2
slope EF  ? slope EF  0
F
G
Sample Problems
N(2, 4)
5. What is the slope of LM & PN?
M(- 4, 2)
Why are they parallel? What is
the slope of MN & LP? Why are
they not parallel? What kind of
quadrilateral is LMNP?
L(- 3, - 1)
2  1
3
slope LM  ? slope LM 
slope PN  ?
slope MN  ?
slope LP  ?
4  3

1
 3
P(4, - 2)
4  2 6

 3 LM || PN why?
2  4 2
42 2 1
slope MN 
 
2  4 6 3
2  1 1
MN not parallel to MN why?
slope LP 

4  3
7
slope PN 
What kind of quadrilateral is LMNP?
Trapezoid
Sample Problems
7. Find the slope of each side and each altitude of  ABC.
A(0, 0) B(7, 3) C(2, - 5) What should you do first?
slope of altitude
from AC = ? 2
5
slope of altitude
from CB = ?  5
8
30 3

70 7
slope AC  ? slope AC  5  0  5
20
2
3  5 8
slope CB  ?
slope CB 

72 5
B (7, 3)
What must be true of the altitude
slope AB  ?
y
3
2
1
A (0, 0)
slope AB 
drawn from the line containing AB?
Altitude must be  to
1 2 3
C (2, -5)
x
the line containing AB.
slope AB  slope of altitude = -1
3
 slope of altitude = -1
7
7
slope of altitude from AB = 3
Sample Problems
9. Identify the legs of right  RST. R(- 3, - 4) S(2, 2) T(14, - 8)
11. Given parallelogram ABCD, A(- 6, - 4) B(4, 2) C(6, 8)
D(- 4, 2). Show that opposite sides are parallel and opposite
sides are congruent.
13. R(- 4, 5) S(- 1, 9) T(7, 3) U(4, - 1) Show that RSTU is a
rectangle. Show that the diagonals are congruent.
Decide what type of quadrilateral HIJK is, then tell why.
15. H(0, 0) I(5, 0) J(7, 9) K(1, 9)
17. H(7, 5) I(8, 3) J(0, - 1) K(- 1, 1)
19. Point N(3, - 4) is on the circle x2 + y2 = 25. Find the slope
of the line tangent to the circle at N.
Section 13-4
Vectors
Homework Pages 541-542:
1-30
Excluding 8, 16
Objectives
A. Understand and utilize the terms
‘vector’, ‘magnitude’, and
‘direction’.
B. Identify and calculate the
horizontal and vertical
components of a vector.
C. Recognize and identify equal, parallel, and perpendicular
vectors.
D. Calculate the magnitude and slope of a vector.
E. Perform vector addition, vector subtraction, and scalar
multiplication.
Vectors
 vector: a line segment with both magnitude and direction,
written as an ordered pair of numbers (H, V) where H is
the horizontal component and V is the vertical component.
y
V
H
x
Vector Components
 horizontal component: the distance traveled left or right
from the starting to the ending point, found by taking
xstop - xstart
 vertical component: the distance traveled up or down from
the starting point to the ending point, found by taking
ystop - ystart
Vector (H, V) = (4, 3)
(7, 8)
V = Stop – Start
V=8–5=3
(3, 5)
H = Stop – Start = 7 – 3 = 4
Vector Terms
• Equal Vectors  vectors with the same magnitude and
direction
– Note that equal vectors may or may not be collinear
• Magnitude of a Vector  The length of the vector.
– Found by using the distance formula
• dot product: (H1, V1)  (H2,V2) = H1H2 + V1V2
 vector addition: (H1, V1) + (H2,V2) = (H1 + H2 , V1 + V2)
 vector subtraction: (H1, V1) - (H2,V2) = (H1 - H2 , V1 - V2)
 scalar multiplication: k(H1,V1) = (kH1, kV1)
2
2
magnitude
:
AB

H

V

V
 slope :
H
Sample Problems
Find AB and AB
3. A(6, 1) B(4, 3)
What should you do first?
‘Order’ of vector dictates that Point A is the
start point and Point B is the stop point.
AB  ?
AB  ( H ,V )
y
3
2
1
B (4, 3)
H = xstop – xstart = 4 – 6 = -2
V = ystop – ystart = 3 – 1 = 2
AB  ( H ,V )  (2, 2)
AB  ?
A (6, 1)
1 2 3
x
AB  H 2  V 2
AB  (2)2  (2)2  4  4  2 2 units
Sample Problems
Perform the scalar multiplication.
11. 3(4, - 1)
The scalar k is 3. The horizontal component H is 4.
The vertical component V is -1.
k(H,V) = (kH, kV)
3(4, -1) = (3 x 4, 3 x -1) = (12, -3)
Sample Problems
Find the vector sum.
21. (3, - 5) + (4, 5)
(H1, V1) + (H2, V2) = ?
(H1, V1) + (H2, V2) = (H1 + H2 , V1 + V2)
u1   H1 ,V1    3, 5 
u2   H 2 ,V2    4,5 
u3  u1  u2   H1 ,V1    H2 ,V2 
u3  u1  u2   H1 ,V1    H2 ,V2    H1  H2 ,V1  V2   3  4, 5  5   7,0
u3  u1  u2  ?
u3  u1  u2   H1 ,V1    H2 ,V2    H1  H2 ,V1 V2   3  4, 5  5   1, 10
Sample Problems – Another look at #21!
Find the vector sum.
21. (3, - 5) + (4, 5)
AB  (3, 5) BC  (4,5)
AC  AB  BC
y
AC  (3, 5)  (4,5)
AC  (3, 5)  (4,5)  (3  4, 5  5)  (7,0)
3
2
1
A(0, 0)
C (7, 0)
1 2 3
B (3, -5)
x
Sample Problems
27. An object, K, is being pulled by two forces KX = (- 1, 5) and
KY = (7, 3). What single force has the same effect as the two forces
acting together? What is the magnitude of this force?
KX = (- 1, 5)
R(6, 8)
y
KY = (7, 3)
X(-1, 5)
Y(6, 8)
3
2
1
K(0, 0)
The result of 2 forces working on an
object at the same time has the SAME
result as the object being worked on by
the first force and THEN being worked
on by the second force.
KR  KX  KY
1 2 3
x KR  (1,5)  (7,3)  (6,8)
KR  H 2  V 2
KR  62  82  36  64  100  10 units
Sample Problems
Find AB and AB
1. A(1, 1) B(5, 4)
5. A(3, 5) B(- 1, 7)
7. A(0, 0) B(5, - 9)
9. A(- 1, - 1) B(- 4, - 7)
Perform the scalar multiplication.
13. 1 6, 9
3
15.  1 (6, - 4)
2
Sample Problems
17. The vectors (8, 6) and (12, k) are parallel. Find k.
19. The vectors (8, k) and (9, 6) are perpendicular. Find k.
Find the vector sum.
23. (- 3, - 3) + (7, 7)
25. (7, 2) + 3(- 1, 0)
Sample Problems
29. M is the midpoint of AB. T is the trisector of AB.
A(2, 3) B(20, 21). Find AB, AM and AT. Find the
coordinates of M & T.
B(20, 21)
M
T
A(2, 3)
Section 13-5
The Midpoint Formula
Homework Pages 545-546:
1-20
Excluding 18
Objectives
A. Understand and utilize the
midpoint formula.
 Theorem 13-5 (Midpoint Formula)
The midpoint (xm , ym) of a segment that joins the points
(x1 , y1) and (x2 , y2) is the point
 x1 + x 2 y1  y 2 
x m , y m   
,
.
2 
 2
(x1 , y1)
(xm , ym)
(x2 , y2)
Sample Problems
Find the coordinates of the midpoint.
What should you do first?
3. (6, - 7) & (- 6, 3)
P2 (-6, 3)
midpoint  ?
 x  x y  y2 
midpoint   1 2 , 1

2
2


y
-1
-2
-3
1 2 3
(0, -2)
x
P1   6, 7  P2   6,3
x1  6, y1  7 x2  6, y2  3
 6  6 7  3 
midpoint  
,
   0, 2 
2 
 2
Is this a REASONABLE answer?
P1 (6, -7)
Sample Problems
Find the length, slope and midpoint of PQ.
7. P(3, - 8) Q(- 5, 2)
What should you do first?
Q (-5, 2)
Label the points: P1 = P = (3, -8), P2 = Q = (-5, 2)
Label the coordinates of the points:
x1 = 3, y1 = -8, x2 = -5, y2 = 2
y
length  distance = ?
-1
-2
-3
slope  ?
y2  y1
slope 
x2  x1
2  8
slope 
5  3
10
5


8
4
x length 
1 2 3
length 
 x 2  x1    y2  y1 
2
 5  3   2  8
2
2
2
 64  100  2 41 units
midpoint  ?
P (3, -8)
 x  x y  y2 
midpoint   1 2 , 1

2
2


 3  5 8  2 
midpoint  
,
   1, 3 
2 
 2
Sample Problems
15. Find the midpoints of the legs and then the length of the
median of the trapezoid CDEF with vertices C(- 4, - 3),
D(- 1, 4), E(4, 4) & F(7, - 3) .
What should you do first?
y
 xC  xD yC  yD 
midpoint CD  ?
midpoint CD  
,

2
2


 4  1 3  4   5 1 
midpoint CD  
,
   , 
2   2 2
 2
E(4,4)
midpoint EF  ?
D(-1,4)
 5 1
 , 
 2 2
3
2
1
 11 1 
 , 
 2 2
1 2 3
C(-4,-3)
 x  xF y E  y F 
mid EF   E
,

2
2


x
F(7,-3)
 4  7 4  3   11 1 
mid EF  
,
 , 
2   2 2
 2
Sample Problems
15. Find the midpoints of the legs and then the length of the
median of the trapezoid CDEF with vertices C(- 4, - 3),
D(- 1, 4), E(4, 4) & F(7, - 3) .
length  distance = ?
y
distance 
 x2  x1    y2  y1 
2
2
D(-1,4)
 5 1
 , 
 2 2
E(4,4)
3
2
1
 11 1 
 , 
 2 2
1 2 3
C(-4,-3)
2
5 1 1
 11
distance         
2 2 2
2
2
2
 16 
     0  8
 2
x
F(7,-3)
2
Sample Problems
Find the coordinates of the midpoint.
1. (0, 2) & (6, 4)
5. (2.3, 3.7) & (1.5, - 2.9)
Find the length, slope and midpoint of PQ.
9. P(- 7, 11) Q(1, - 4)
M is the midpoint of AB. Find the coordinates of B.
11. A(1, - 3) M(5, 1)
13. A(0, 0) & B(8, 4), show P(2, 6) is on the perpendicular
bisector of AB.
Sample Problems
17. Show that OQ & PR have the same midpoint. What kind
of quadrilateral is OPQR? Show that the opposite sides
are parallel. Show the opposite sides are congruent.
P(2, 6)
Q(9, 9)
R(7, 3)
O
Sample Problems
19. In right OAT M is the midpoint of AT. What are the
coordinates of M? Find MA, MT & MO. Find the
equation of the circle that circumscribes the triangle.
A(0, 8)
M
T(- 6, 0)
O
Section 13-6
Graphing Linear Equations
Homework Pages 550-551:
1-33 ALL
Excluding 1, 3, 32
Objectives
A. Identify linear equations.
B. Understand and utilize the
standard form of a linear equation.
C. Understand and utilize the slopeintercept form of a linear
equation.
D. Understand and apply multiple methods for solving systems of
linear equations.
Remember!
A linear equation is any equation whose solution graph is a
line.
 Methods for Graphing a Line:
1. Plot two or more points
Remember  any two points are contained on one unique line.
For graphing purposes, I recommend you plot at least 3 points
•
•
y  2x 1
If x = ?
y
Then y =
0
1
1
3
2
5
3
2
1
1 2 3
x
 Theorem 13-6 (Standard Form of a Linear Equation)
The graph of any equation that can be written in form Ax + By = C
where A is zero or positive, A and B are not both zero, and A, B
and C are integers, is a line. Therefore, Ax + By = C is the
standard form of a linear equation.
A
slope  
B
C 
x - intercept is the point  ,0 
A 
 C
y - intercept is the point  0, 
 B
 Theorem 13-7 (Slope Intercept Form of a Line)
A line with the equation y = mx + b has slope m and
y-intercept b.
 Methods for Graphing a Line:
1. Plot two points
2. Plot one point (Y-intercept) and rise and run according to
the slope.
y  2x 1
Slope-intercept
form of a linear
equation is?
y = mx + b
m  2 or m 
2
1
b=1
y-intercept = (0, 1)
y
3
2
1
1 2 3
x
 Solving Systems of Linear Equations - Graphing
• Graph the two linear equations and identify the intersection
point.
– The point of intersection of the two lines gives the x and
y coordinates that will make BOTH linear equations
true.
– Can be used on any system but your answer is only as
accurate as your graph.
 Solving Systems of Linear Equations - Graphing
y  x 1
y  x 1
y
Point of intersection (-1, 0)
CHECK your answer!
y  x  1  (1)  1  0
y   x  1  (1)  1  0
3
2
1
1 2 3
x
Solving Systems of Linear Equations –
Isolate and Substitute
• Solve for (isolate) x or y in one equation.
• Substitute the expression from the step above into the
second equation.
• Solve the second equation for the remaining variable.
• Substitute the answer from the step above into either
original equation and solve for the other variable.
• Can be used on any system but works best when one
coefficient is either 1 or - 1.
Solving Systems of Linear Equations –
Isolate and Substitute
2 x  y  12
3 x  2 y  17
Solution (1, 10)
Isolate (solve for) y in the first equation  y = 12 – 2x
Using the result of the isolation, substitute for y in the
second equation  3x – 2(12 – 2x) = -17.
Solve for x  3x – 24 + 4x = -17
7x = 7  x = 1
Substitute x value back into original equation 
2(1) + y = 12  y = 10
Solving Systems of Linear Equations –
Linear Combination
• Addition: add the two equations to eliminate one of the
variables. Works only if the coefficients of one of the
variables are opposites.
• Subtraction: subtract the two equations to eliminate one of
the variables. Works only if the coefficients of one of the
variables are the same.
• Multiply one or both equations by a constant in order to
create coefficients of the same variable that are either the
same or opposites, then add/subtract.
Solving Systems of Linear Equations –
Linear Combination
5 x  2 y  16
3 x  4 y  6
5(2)  2 y  16
Solution (2, 3)
2y  6
y 3
What to do?
Graphing would be complicated.
Isolating x or y would leave nasty fractions.
Adding or subtracting equations will not eliminate variable.
Multiply both sides of first equation by 2  10x + 4y = 32
Now add the two equations:
10 x  4 y  32
3 x  4 y  6

13x  0 y  26
x2
Sample Problems
Find the x & y intercepts and the slope, then graph.
7. 3x + y = - 21
How do you find the x-intercepts for a solution graph?
3x + 0 = -21
x = -7
3(0) + y = -21
y = -21
Ax + By = C
m
A
3
m    3
B
1
y = -3x - 21
Plug in zero for the y-value and solve for the x-value.
x-intercept = (-7, 0)
How do you find the y-intercepts for a solution graph?
Plug in zero for the x-value and solve for the y-value.
y-intercept = (0, -21)
How do you find the slope of the solution graph?
The equation 3x + y = -21 is in which form?
The equation 3x + y = -21 is in standard form.
What is the standard form of a linear equation?
where A is positive or zero, A and B are not both zero,
and A, B, and C are integers.
Slope of the line when using standard form = ?
Any other way of finding the slope?
Convert the equation into slope-intercept form!
Sample Problems
Find the x & y intercepts and the slope, then graph.
7. 3x + y = - 21
x-intercept = (-7, 0)
Graph the solution.
y-intercept = (0, -21)
(-7, 0)
3
m    3
1
y
-3
-6
-9
3 6 9
(0, -21)
x
Sample Problems
Solve the system
Solution (2, -3)
29. 4x + 5y = - 7
2x - 3y = 13  Multiply this equation by 2  4x – 6y = 26
What are the 3 methods you can use to solve systems of linear equations?
1. Graph
2. Isolate and substitute
3. Linear combination
Which of these would you use?
Why would you NOT use the graphing method for this problem?
Linear combination will be the method you use most frequently!
What would you do in order to do linear combination to this system?
4x + 5y = - 7
4x + 5(-3) = - 7
- (4x - 6y = 26)
Plug back into equation!
4x – 15 = -7
----------------4x = 8
0x + 11y = -33
x=2
y = -3
Sample Problems
1 1
1. Graph y = mx if m = 2, - 2, , 2 2
3. Graph y  1 x  b for b = 0, 2, - 2, - 4
2
5. Graph y = 0, y = 3, y = - 3
Find the x & y intercepts and the slope, then graph.
9. 3x + 2y = 12
11. 5x + 8y = 20
Sample Problems
Find the slope, x & y intercepts, then graph.
13. y = 2x - 3
15. y = - 4x
2
17. y   x  4
3
19. 4x + y = 10
21. 5x - 2y = 10
23. x - 4y = 6
Sample Problems
Solve the system
25. x + y = 3
x-y=-1
27. x + 2y = 10
3x - 2y = 6
Section 13-7
Writing Linear Equations
Homework Page 555:
1-28
Excluding 6, 16
Final Answers must be in STANDARD
FORM for a linear equation (Ax + By = C)
Objectives
A. Understand and utilize the pointslope form of a linear equation.
B. Use various pieces of information
about a line or linear equation to
determine the standard, slopeintercept, and/or point-slope form
of the linear equation.
 Theorem 13-8 (Point Slope Form of a Line)
An equation of the line that passes through the point (x1, y1)
and has slope m is y - y1 = m(x - x1).
 Writing an Equation of the Line
• Find the slope and one point on the line.
• Use the point-slope form of a line; put the slope in for m
and the point in for (x1 , y1).
• Distribute and rearrange the equation until it is in standard
form, with all coefficients being integers.
Sample Problems
Write the equation of the line in standard form.
7
5. slope =  y-intercept (0, 8)
5
What is the point-slope form of a linear equation: y  y1  m  x  x1 
Fill in the given point and slope:
Put in standard form (Ax + By = C):
7
y  8    x  0
5
7
y 8   x
5
7
x  1y  8
5
7 x  5 y  40
Sample Problems
25. Write the standard form of the equation of a line through (5, 7)
and parallel to the line y = 3x – 4.
y = 3x – 4 is in what form?
What is the slope-intercept form of a linear equation?
y = mx + b where m is the slope and b is the y-intercept.
What is be the slope of the line y = 3x - 4?
What will be the slope of the ‘new’ line?
If the lines are to be parallel, then, m1 = m2. So m2 = 3.
You are given the point (5, 7).
What is the point-slope formula?
Fill in what you know.
Put in standard form.
m1 = 3
y  y1  m  x  x1 
y  7  3  x  5
y  7  3 x  15
3x  y  8
Sample Problems
Write the equation of the line in standard form.
1. slope = 2 y-intercept = (0, 5)
1
3. slope =
y-intercept (0, - 8)
2
5. slope =  7 y-intercept (0, 8)
5
7. x-intercept (8, 0) y-intercept (0, 2)
9. x-intercept (- 8, 0) y-intercept (0, 4)
11. point (1, 2) slope = 5
1
13. point (- 3, 5) slope =
3
Sample Problems
15. point (- 4, 0) slope =
1

2
17. line through (1, 1) & (4, 7)
19. line through (- 3, 1) & (3, 3)
21. vertical line through (2, - 5)
23. line through (5, - 3) and parallel to the line x = 4
27. line through (- 3, - 2) and perpendicular to the line 8x - 5y = 0
29. perpendicular bisector of the segment joining (0, 0) & (10, 6)
31. the line through (5, 5) that makes a 45° angle measured
counterclockwise from the positive x axis
Section 13-8
Organizing Coordinate Proofs
Homework Pages 558-559:
1-10
Draw and label each diagram on
graph paper!
Objectives
A. Understand and apply the term
“coordinate proof”.
B. Given a polygon, choose an
appropriate placement of
coordinate axes on the polygon to
assist in a coordinate proof.
C. Given a polygon properly placed on a coordinate plane, choose
appropriate coordinates (variable and fixed) of the vertices of the
polygon to assist in a coordinate proof.
Coordinate Proofs
A coordinate proof:
– Is a method of proving a conditional statement
– Is similar to the direct and indirect proof methods you have already
encountered
– Has all of the requirements of any other proof method
• Identify the conditional statement to be proved
• Identify the given information and diagrams
• Identify the proof statement
• Provide logical steps from given information to proof
statement
– Can be used in conjunction with a direct or indirect proof.
– Uses objects placed in a coordinate plane to assist with the proof
– Uses variable coordinates from the coordinate plane to provide a
GENERAL proof of the conditional statement.
Coordinate Proof: Placing the Object on the Coordinate Plane
Consider placing an isosceles right triangle on a coordinate
plane. What are the characteristics of an isosceles right
triangle?
– Polygon with 3 sides
– Has exactly one right angle
– Has congruent legs
Coordinate Proof: Placing the Object on the Coordinate Plane
Placing an isosceles right triangle on a coordinate plane.
Does this appear to be a good
placement of the triangle in the
coordinate plane? Why or why not?
y
3
2
1
x
1
2
3
Is it easy to prove there is a right
angle in the triangle?
Is it easy to prove there are congruent
legs in the triangle?
Are the coordinates of the vertices of
this triangle fixed real numbers?
If we wanted to generalize the proof,
how many variable coordinate would
be required?
Would calculation of the slopes of the
lines be an easy task?
Would calculation of the lengths of the
sides be an easy task?
Coordinate Proof: Placing the Object on the Coordinate Plane
Placing an isosceles right triangle on a coordinate plane.
Can we make our life easier?
Is there any rule that disallows us
from rotating the triangle on the
coordinate plane?
NO! So choose wisely!
y
3
2
1
x
1
2
3
Coordinate Proof: Placing the Object on the Coordinate Plane
Placing an isosceles right triangle on a coordinate plane.
Is it easy to prove there is a right
angle in the triangle?
y
(0, a)
?)
x
(0, 0)
?)
What is the x-coordinate of this point?
What is the y-coordinate of this point?
Be sure that it is useful in a general
proof!
What is the y-coordinate of this point?
What is the x-coordinate of this point?
Be sure that it is useful in a general
proof and it makes an isosceles
triangle!
What is the x-coordinate of this point?
(?, 0)
(a,
What is the y-coordinate of this point?
Coordinate Proof: Placing the Object on the Coordinate Plane
Placing an isosceles right triangle on a coordinate plane.
y
(0, a)
x
(0, 0)
(a, 0)
Is it easy to prove there are congruent
legs in the triangle?
Are the coordinates of the vertices of
this triangle fixed real numbers?
How many variable coordinate would
be required?
Would calculation of the slopes of the
lines be an easy task?
Would calculation of the lengths of the
sides be an easy task?
Coordinate Proof: Placing the Object on the Coordinate Plane
Steps to properly placing a figure on a coordinate plane to
assist in a coordinate proof:
1. If one or more right angles exist in the figure, place one of
them at the intersection of the coordinate axes (origin).
2. If one or more sets of parallel lines exist in the figure,
place at least one of the parallel sides on either the x-axis
or the y-axis.
3. In MOST figures, it is best to use the origin as one of the
vertices of the figure.
4. Whenever possible, place other vertices on the
x-axis (x, 0) or on the y-axis (0, y).
Organizing Coordinate Proofs
Some Sample Diagrams
(0,
(?,b)
?)
(?, 0)
?)
(0,
(?, ?)
(b,
c)
scalene right triangle
(?,
(a, ?)
0)
(?, ?)
(0,
0)
(?, ?)
(0,
b)
(?, ?)
(a,
0)
scalene triangle
scalene triangle
(?, ?)
(c,
0)
(?,0)
?)
(a,
Organizing Coordinate Proofs
isosceles triangle
isosceles triangle
(?, ?)
(a,
b)
(?, ?)
(0,
b)
(?, ?)0)
(-a,
(?, ?)
(a,
0)
(0,
(?,0)
?)
(2a,
(?, ?)
0)
(?, ?)
(b,
c)
(?, ?)
(0,
a)
(a(?,
+ b,
?) c)
(?, ?)
(b,
a)
(?, ?)
(0,
0)
(?, ?)
(b,
0)
rectangle
(0,
(?,0)
?)
(?, ?)
(a,
0)
parallelogram
Organizing Coordinate Proofs
(?, ?)
(b,
c)
(0,
(?,0)
?)
(?, ?)
(b,
c)
(?,-?)b, c)
(a
(?, ?)
(a,
0)
isosceles trapezoid
(?, ?)
(d,
c)
(0,
(?,0)
?)
(?, ?)
(a,
0)
trapezoid
Sample Problems
Supply the missing coordinates without using any new variables.
(a, b)
(?, ?)
(?, ?)
(- f, 0)
1. rectangle
(m, n)
(?, ?)
(?, ?)
(f, 0)
3. square
(?, ?)
(?, ?)
(h, 0)
5. parallelogram
(s, 0)
7. equilateral triangle
Sample Problems
(?, b)
(?, ?)
(c, 0)
9. rhombus
Section 13-9
Coordinate Geometry Proofs
Homework Page 562:
1-10
Make sure you do two-column proofs
and include a properly annotated
diagram!
Objectives
A. Prove conditional statements by
using coordinate geometry proof
methods.
Writing Coordinate Proofs
• To write a coordinate proof of a geometric theorem the
first step is to place the diagram on a generic graph i.e. a
graph without numbers.
• The second step is to label each vertex with variable
coordinates. Each x-coordinate and each y-coordinate
must be assigned a different variable unless a relationship
has already been proven to exist.
– To minimize the number of letters used to label the
vertices, it is wise to place one vertex on the origin and
as many other vertices as possible on the coordinate
axes.
Writing Coordinate Proofs
• The third step is to write the given and the proof statements
as algebraic expressions, using the coordinates from the
diagram.
• The fourth step is to create the body of the proof by using
the rules of algebra to transform the given equation into the
equation in the prove statement.
– Write as two-column proofs.
– Most proofs can be accomplished by using the
equations from theorems 13-1 to 13-7 along with the
definitions of slope and vectors to write or transform
your given statement.
Coordinate Geometry Proofs - Example
Prove that the diagonals of an isosceles trapezoid are congruent.
First step?
Diagram on a generic graph (without numbers).
Second step?
y
The second step is to label each vertex with
variable coordinates. Each x-coordinate and
each y-coordinate must be assigned a different
variable unless a relationship has already been
proven to exist.
R (0, 0)
U (n, 0)
x
Third step?
The third step is to write the given and the proof statements as algebraic
expressions, using the coordinates from the diagram.
How would you show these diagonals to be congruent? Distance formula?
S (p, q)
d
T (n - p, q)
 x2  x1    y2  y1 
2
2
RT 
 n  p  0   q  0
US 
n  p  0  q
2
2
2

Since RT = US, the diagonals are congruent.
2

n  p
n  p
2
2
 q2
 q2
Coordinate Geometry Proofs - Example
Prove that the diagonals of an isosceles trapezoid are congruent.
The fourth step is to create the body of the
Fourth step?
proof by using the rules of algebra to
y
transform the given equation into the equation
S (p, q)
T (n - p, q)
in the prove statement.
R (0, 0)
U (n, 0)
x
1. Quadrilateral RSTU; ST || RU;
1. Given
RS  TU; diagram with coordinates given.
2. d 
 x2  x1    y2  y1 
2
2
2. The distance d between
points (x1 , y1 ) and (x 2 , y 2 ) is given by
d
 x2  x1    y2  y1 
2
2
Coordinate Geometry Proofs - Example
Prove that the diagonals of an isosceles trapezoid are congruent.
The fourth step is to create the body of the
Fourth step?
proof by using the rules of algebra to
y
transform the given equation into the equation
S (p, q)
T (n - p, q)
in the prove statement.
R (0, 0)
U (n, 0)
3. RT 
 n  p  0   q  0
4. US 
n  p  0  q 
2
2
2
2

x

n  p
n  p
2
2
 q2
3. Substitution and simple math.
 q2
4. Substitution and simple math.
5.US  RT
5. Substitution.
6.US  RT
6. Definition of congruence
Sample Problems
Prove that the segment joining the midpoints of the legs of a trapezoid is
parallel to the bases and has a length equal to half the sum of the lengths
of the bases.
y
What might be useful to add to the diagram?
N (b, c)
M (d, c)
E
O
What is the midpoint formula?
 x1  x2 y1  y2 
midpoint  
,

2
2


F
P (a, 0)
x
What are the coordinates of point E?
E  midpoint ON
b0 c0 b c 

,
 , 
2  2 2
 2
What are the coordinates of point F?
F  midpoint MP
d a c0 d a c 

,
, 

2   2 2
 2
Sample Problems
Prove that the segment joining the midpoints of the legs of a trapezoid is
parallel to the bases and has a length equal to half the sum of the lengths
of the bases.
b c
d a c
E ,  F
, 
2 2
 2 2
y
N (b, c)
E
F
O
P (a, 0)
mEF  ?
mOP
M (d, c)
x
How would you prove EF || OP || MN ?
What is the slope formula? m  y2  y1
x2  x1
c c

0
2
2
mEF 

0
d  a b d  a b
mMN  ?

2
2
2
cc
0
0

0
0
mMN 

0
?
mOP 
 0
d b d b
a0 a
Sample Problems
Use coordinate geometry to prove each statement. First draw a figure and
choose convenient axes and coordinates.
1. The diagonals of a rectangle are congruent. (Theorem 5-12)
Sample Problems
Use coordinate geometry to prove each statement. First draw a figure and
choose convenient axes and coordinates.
3. The diagonals of a rhombus are perpendicular. (Theorem 5-13) (Hint:
Let the vertices be (0, 0), (a, 0), (a + b, c), and (b, c).
Show that c2 = a2 – b2.)
Sample Problems
5. Prove that the segment joining the midpoints of the diagonals of a
trapezoid is parallel to the bases and has a length equal to half the
difference of the lengths of the bases.
y
N (b, c)
O
M (d, c)
P (a, 0)
x
Sample Problems
7. Prove that the figure formed by joining, in order, the midpoints of the
sides of quadrilateral ROST is a parallelogram.
y
R (2b, 2c)
T (2d, 2e)
O
S (2a, 0)
x
Sample Problems
9. Prove that the angle inscribed in a semicircle is a right angle. (Hint:
The coordinates of C must satisfy the equation of a circle.)
Chapter Thirteen
Coordinate Geometry
Review
Homework Page 568: 1-18