Unit 5 Radicals and Equations of a Line

1
Unit 5 Radicals and Equations of a Line

• This unit has a review section on
  solving/reducing radicals, and the laws of
  exponents.
• This is required to solve the Pythagorean
  Theorem.
• This unit also covers the three standard equation
  of a line used to graph in the coordinate plane
  (Standard equation, slope-intercept, and
  point-slope).
• The distance formula, midpoint, endpoint, and
  slope are all addressed in the coordinate plane,
  as well as parallel and perpendicular lines.

2
Standards

• SPIs taught in Unit 5
• SPI 3108.1.4 Use definitions, basic postulates,
  and theorems about points, lines, angles, and
  planes to write/complete proofs and/or to solve
  problems.
• SPI 3108.3.1 Use algebra and coordinate geometry
  to analyze and solve problems about geometric
  figures (including circles).
• SPI 3108.3.2 Use coordinate geometry to prove
  characteristics of polygonal figures.
• CLE (Course Level Expectations) found in Unit 5
• CLE 3108.1.4 Move flexibly between multiple
  representations (contextual, physical written,
  verbal, iconic/pictorial, graphical, tabular, and
  symbolic), to solve problems, to model
  mathematical ideas, and to communicate solution
  strategies.
• CLE 3108.1.7 Use technologies appropriately to
  develop understanding of abstract mathematical
  ideas, to facilitate problem solving, and to
  produce accurate and reliable models.
• CLE3108.2.3 Establish an ability to estimate,
  select appropriate units, evaluate accuracy of
  calculations and approximate error in measurement
  in geometric settings.
• CLE 3108.3.1 Use analytic geometry tools to
  explore geometric problems involving parallel and
  perpendicular lines, circles, and special points
  of polygons.
• CFU (Checks for Understanding) applied to Unit 5
• 3108.1.1 Check solutions after making reasonable
  estimates in appropriate units of quantities
  encountered in contextual situations.
• 3108.1.2 Determine position using spatial sense
  with two and three-dimensional coordinate
  systems.
• 3108.1.3 Comprehend the concept of length on the
  number line.
• 3108.1.5 Use technology, hands-on activities, and
  manipulatives to develop the language and the
  concepts of geometry, including specialized
  vocabulary (e.g. graphing calculators,
  interactive geometry software such as Geometers
  Sketchpad and Cabri, algebra tiles, pattern
  blocks, tessellation tiles, MIRAs, mirrors,
  spinners, geoboards, conic section models, volume
  demonstration kits, Polydrons, measurement tools,
  compasses, PentaBlocks, pentominoes, cubes,
  tangrams).
• 3108.1.7 Recognize the capabilities and the
  limitations of calculators and computers in
  solving problems.
• 3108.2.6 Analyze precision, accuracy, and
  approximate error in measurement situations.
• 3108.3.1 Prove two lines are parallel,
  perpendicular, or oblique using coordinate
  geometry.
• 3108.3.2 Connect coordinate geometry to geometric
  figures in the plane (e.g. midpoints, distance
  formula, slope, and polygons).
• 3108.3.4 Apply the midpoint and distance formulas
  to points and segments to find midpoints,
  distances, and missing information in two and
  three dimensions.

3
Reducing a Radical

• There are three conditions that must be met for a
  radical expression to be in its simplest form
• The number under the radical sign has no perfect
  square factors other than 1
• The number under the radical sign does not
  contain a fraction
• A denominator does not contain a radical
  expression

4
Reducing the Square Root of a Fraction

• Reduce this Square Root v4/3
• We can rewrite this v4 /v3
• The square root of 4 is 2, so we now have 2/v3.
• We cannot stop here, there is a radical in the
  denominator.
• Therefore, we multiply the top and bottom by top
  and bottom by v3/v3 ( a form of 1)
• So( 2 v3) / ( v3 v3) 2 v3 / 3

5
Reducing a Radical with a Perfect Square

• Reduce this radical v24/ v3
• The square root of 24 can be rewritten as (v4
  v6)/ v3
• We can solve the square root of four (it is 2)
  and now have (2 v6)/ v3
• The square root of 6 can be rewritten as the
  square root of 2 times the square root of 3,
  which gives us ( 2 v2 v3)/ v3
• We can cancel out the square roots of 3, and now
  we have 2 v2

6
A Deeper Look at Reducing Radicals

• Ultimately, to reduce radicals we look for
  perfect squares
• For example, reduce this v75
• This has one perfect square in it 25
• Therefore, we have 253, which is 553
• Therefore we can reduce this to 5 v3
• There are three ways to reduce a radical

7
How to Reduce a Radical

• Recognize perfect squares. If you know your
  perfect squares up to about 13, you can often
  reduce a radical in your head (like reducing v75)
• The Factor Tree. Whee ?
• Using a Calculator
• What A calculator? How????

8
Recognize the Square

• Reduce this v60
• This is 4 x 15. Therefore it reduces to 2 v15
• Reduce this v45
• This is 9 x 5. Therefore it reduces to 3 v5
• Reduce this v90
• This is 9 x 10. Therefore it reduces to 3 v10
• This is a good method for simple radicals

9
Factor Tree

• Reduce this v80
• The factor tree requires you to reduce a number
  to its prime numbers
• 80 is 8 x 10, which is 2 x 2 x 2 x 2 x 5
• Therefore, I can pull out the 2s and leave the
  five
• This gives me 4 v5
• Most of you remember how to do this

10
Calculator

• Lets look at this one again v80
• Again, Ill look for perfect squares, but this
  time using the calculator
• Go to the y button (top left)
• Enter this 80/x2 (use the x,t,Ø,n button for
  X, and then use the squared button)
• This forces the calculator to divide 80 by every
  single rational number from 1 to 80

11
Calculating the Square

• The calculator first assigns X the value of 1.
  Therefore Y is 80 (80/12)
• It then continues up in infinite increments
  basically forever
• What we want to do is identify the largest value
  of X, and whatever is left over will still be
  under the radical sign

12
The Table

• After we enter 80/x2, we need to enter this
• Press the 2nd button, then table F5 (top
  right button)
• Here you will see an X / Y chart
• What we focus on is the Y column
• We need to find the smallest value of Y, where Y
  is a whole number greater than 1 (1 cant be a
  perfect square, right?)

13
Analyzing the Table

• As we look at the table, we see that the smallest
  whole number in the Y table is 5
• We see the X value is 4
• We can therefore write this 4 v5
• Remember, the value of Y is what is left under
  the radical sign
• If we cannot find a whole number for Y, we cannot
  reduce the fraction

14
Try Another

• Reduce this using the Calculator v135
• Go to the y button (you may have to hit
  clear to remove your last problem) and enter
  135/X2
• Now hit 2nd and Table
• We look for the smallest whole value of Y, and
  find 15. The matching X value is 3
• So we write 3 v15
• To confirm this is right, we would multiply 3 x 3
  x 15, and we get 135.

15
Example

• You try
• v175
• Answer 5 v7
• v294
• Answer 7 v6
• v168
• Answer 2 v42
• v270
• Answer 3 v30

16
Assignment

• Page 399 1-15
• 3 Radicals Worksheets

17
Unit 5 Quiz 1Reduce all radicals to their
simplest form

• Answers

1. v156
2. v240
3. v340
4. v76
5. v40
6. v150
7. v196
8. v90
9. v220
10. v540

1. 2 v39
2. 4 v15
3. 2 v85
4. 2 v19
5. 2 v10
6. 5 v6
7. 14
8. 3 v10
9. 2 v55
10. 6 v15

18
Unit 5 Quiz 2Reduce all radicals to their
simplest form

• Answers

1. (v156) (v72)
2. (v240) (v15)
3. (v340) / 2
4. (v76) / (v19)
5. (v40) (v20)
6. (v150) (v24)
7. (v196) (v36)
8. (v90) / 6
9. v(240/10)
10. (v540) (4v15)

1. 2v39 - 6v2
2. 5v15
3. v85
4. 2
5. 20v2
6. 3v6
7. 8
8. (v10) / 2
9. 2v6
10. 10v15

19
Multiplying with Exponents

• A quick look at exponents
• When multiplying the same base number with
  exponents, add the exponent
• For example 53 x 52 55
• This is the same as writing this
• 51 x 51 x 51 x 51 x 51 which equals 55

53 52
20
Multiplying with Exponents

• What is y3 x y3
• it would be y6
• What is ya x yb
• it would be yab
• What is m6 x m-4
• it would be m2
• What is n2 x n3 x n
• it would be n6

21
Parenthesis and Exponents

• When we have parenthesis, we multiply the outer
  exponent times the inner exponent
• For example (x4)3 x12
• This is because we are really doing this
• (x4) x (x4) x (x4)
• And we know that when multiplying the same base
  number, we add exponents
• Thus we add the exponents 4 4 4 12
• Which is how we get x12

22
Parenthesis and Exponents

• What is (x5)3
• It would be x15
• What is (5x2)2
• It would be 52x4
• What is (52x3)-2
• It would be 5-4x-6
• What is (3-3y-4)-2
• It would be 36y8

23
Dividing with Exponents

• Dividing base numbers with exponents
• When we divide the same base number with
  exponents, we take the top exponent minus the
  bottom exponent
• For example
• X5/X3 X2
• An example with real numbers 53/52 5
• This is 125/25 5, so we have proved this to be
  true

24
Dividing with Exponents

• What is 105/102
• It would be 103
• What is x7/x3
• It would be x4
• What is 53x5/5x2
• It would be 52x3
• What is 43y2/45y7
• It would be 4-2y-5

25
Creating Positive Exponents

• Negative exponents are not considered
  simplified
• A proper answer in math will only have positive
  exponents
• To turn a negative exponent into a positive
  exponent, put it under a dividing line
• For example 10-2 1/102
• This can be rewritten as 1/100
• Here is how it works

26
There is a Pattern

• 103 1000
• 102 100
• 101 10
• 100 1
• 10-1 1/101
• 10-2 1/102 or 1/100
• 10-3 1/103 or 1/1000
• So we simply put the base number under the
  divider line and turn the negative exponent into
  a positive exponent

27
Converting Negative Exponents

• What if the exponent is already under the
  dividing bar, and is negative?
• For example 5/x-2
• In this case, we move the number above the
  dividing bar and make it positive 5x2
• So our rule will be this If we have a negative
  exponent, go to the opposite side of the dividing
  bar, and rewrite the exponent as a positive
  exponent

28
Converting Negative Exponents

• What is x-3
• It would be 1/x3
• What is 5-2y-6
• It would be 1/52y6
• What is 4-3y4/x2z-3
• It would be y4z3/43x2 Here we moved the 43 under
  the dividing line, and the z3 above the dividing
  line, and left the other two variables where they
  were
• What is 4-3/4-5
• It would be 45/43, or 45-3 or 42

29
The Zero Exponent

• ANYTHING TO THE ZERO POWER IS ONE.
• 1
• ONE
• 1!
• Get it?!!!
• For example 50 1, or x0 1

30
Solving the Zero Exponent

• What is 5a0
• It would be 5 (because 5 x 1 5)
• What is 5ab0
• It would be 5a (because 5 x a x 1 5a)
• What is (45810x9y2)0
• It would be 1 (that whole expression is to the
  zero power, so the whole thing is 1)
• What is (2343y8)0 x (1210y8z9)0
• It would be 1 x 1, which is of course 1.

31
Summary

• Multiplying base numbers Add exponent
• Dividing base numbers Subtract bottom exponent
  from top exponent
• Exponents in Parenthesis Multiply outside
  exponent times inside exponent
• Negative Exponents Move expression to opposite
  side of dividing line, convert exponent from
  negative to positive
• Exponent 0, the entire expression which is
  raised to the zero power 1

32
Unit 5 Quiz 2aReduce all radicals to their
simplest form

• Answers

1. (v156) (v72)
2. (v240) (v15)
3. (v340) / 2
4. (v76) / (v19)
5. (v40) (v20)
6. (v150) (v24)
7. (v196) (v36)
8. (v90) / 6
9. v(240/10)
10. (v540) (4v15)

1. 2v39 - 6v2
2. 5v15
3. v85
4. 2
5. 20v2
6. 3v6
7. 8
8. (v10) / 2
9. 2v6
10. 10v15

33
Unit 5 Quiz 3

• When multiplying numbers with the same base, you
  ________ exponents
• When dividing numbers with the same base, you
  _______ the bottom exponent from the top exponent
• When solving exponents in parenthesis, you
  ________ the outer exponent times the inner
  exponent
• When solving negative exponents, you _______ the
  base number, and make the exponent positive
• Any number with an exponent of zero __________
• Reduce (leave in exponent form) m6 x m-4
  ________
• Reduce (leave in exponent form) (5x2)2
  ____________
• Reduce (leave in exponent form) x7/x3
  _____________
• Reduce (leave in exponent form) 5-2y-6
  _____________
• Reduce (2343y8)0 x (1210y8z9)0
  _________________

34
Reducing Radicals with Exponents

• Example

• Answer

1. v13
2. v13 x v13
3. (v13)2
4. va
5. (va)2
6. va2
7. va3 v a2 x a
8. va4 v a2 x a2
9. va5 v a4 x a
10. va6 v a3 x a3

1. v13
2. 13
3. 13
4. va
5. a
6. a
7. a va
8. a2
9. a2 va
10. a3

35
Assignment

• Page 829 1-20
• Exponents Handout
• Another Exponents Handout

36
Pythagorean Theorem (500 BC)

• It was believed that Pythagoras discovered this
  theorem when waiting for the tyrannical ruler,
  Polycrates.
• While looking at the floors square tiling of the
  palace of Polycrates, Pythagoras thought of this
  interesting idea A diagonal line may be used to
  cut or divide the square, and two right triangles
  would be produced from the cut sides.
• Other stories tell us that he then spent time on
  the sand at a beach drawing diagrams until he
  came up with the final equation

37
The Distance Formula

• Background The distance formula is based upon
  the Pythagorean Theorem
• The Pythagorean Theorem states that in a right
  triangle, where you have side A, side B, and side
  C, you can calculate a side thusly a2 b2 c2,
  where c is the hypotenuse, or the longest side of
  the right triangle

38
Distance Formula

• You can draw any line on a graph, and then create
  a right triangle with that line as the hypotenuse

And
So
On
39
Distance Formula

• Remember, to find the length of the hypotenuse
  (the long side C) we need to know the length of
  the two short sides (side A and side B)
• We can easily calculate that it would be the big
  number minus the small number just like on a
  ruler
• So the length of side A is the big x number minus
  the small x number 4 (-4) 8
• The length of side B is the big Y number minus
  the small Y number 5 (-2) 7

(4,5)
C
B
(-4,-2)
(4,-2)
A
40
Distance Formula

• From the previous slide, we know that side A is 8
  units long, and side B is 7 units long
• We also know that A2 B2 C2
• So we have 82 72 C2
• Therefore to find out how long C is, we have to
  take the square root of both sides
• vC2 v(82 72)
• Or C v(82 72)

(4,5)
C
B
(-4,-2)
(4,-2)
A
So C 10.63
41
Distance Formula-I will ask this equation on a
test

• Using the previous example, we can come up with a
  formula which will work to find the distance of
  any line
• Instead of C , we will D (for distance)
• d v(x2-x1)2 (y2-y1)2
• Remember, this is big x minus small x, and then
  square it, and big y minus small y, and then
  square it, add those two squares, then take the
  square root of that sum

42
Examples

• Given point a (5,2) and point b (-4,-1) find the
  distance between the two points
• D v(5-(-4))2 (2 (-1))2
• D v(9)2 (3)2
• D v(81 9)
• D v90
• D 9.48

Would it matter if you subtracted the numbers the
other way around? Hmm D v(-4 5)2 (-1
-2)2 D v(-9)2 (-3)2 D v(81 9 D v(90)
no, it does not D 9.48 matter
43
Midpoint

• If you wanted to know the halfway point between
  two numbers, you would add them together, and
  divide by two.
• For example, what is halfway between 1 and 9?
• 1 9 10, then divide by 2, or 10/2 5
• Therefore, 5 is the midpoint of 1 and 9

44
Midpoint Formula

• If we have segment AB, with the coordinates of A
  (x1, y1) and the coordinates of B (x2, y2)
  all we do is add the xs and divide by two, and
  add the ys and divide by two.
• We will call the midpoint M.
• Therefore, to find Mx (x1 x2)/2, My
  (y1 y2)/2

45
Example

• Segment QS has endpoints Q (3,5) and S (7,-9)
• What is the midpoint M?
• X coordinate of M (37)/2 5
• Y coordinate of M (5 -9)/2 -2
• The coordinates for midpoint M (5,-2)
• By the way. I have this loaded on my calculator.
  Youre welcome to have it. ?

46
Finding an Endpoint

• What if you have an endpoint, and a midpoint, and
  want the other endpoint?
• The midpoint of Segment AB (3,4)
• Endpoint A (-3,-2)
• What is Endpoint B?
• Just substitute into the midpoint formula
• Solving for X2, we have 3 (-3 X2)/2
• Therefore, -3 X2 6, and X2 9
• Solving for Y2, we have 4 (-2 Y2)/2
• Therefore, -2 Y2 8, and Y2 10
• The coordinates for B (9,10)

47
Assignment

• Page 54 6-44
• Worksheet 1-6
• Pythagorean Theorem Word Problems Handout
• Pythagorean Word Problems 2

48
Unit 5 Quiz 4

1. Problems 1 through 5 Graph points A,B,C,D,E
   (Hand draw one graph with five points label
   each)
2. Problems 6 through 10 Use the same points
   Calculate the midpoint between the two points in
   each set show answers below graph
3. Problems 11 through 15 Use the same points
   Calculate the distance between the two points in
   each set show answers below graph

• 6, 11. A (1,4) B (3,-5)
• 6 (midpoint) 11 (distance)
• 7, 12. C (0,1) D (2, 6)
• 7 (midpoint) 12 (distance)
• 8, 13. E (0,0) F (4,6)
• 8 (midpoint) 13 (distance)
• 9, 14. G (-3,-4) H (-2,7)
• 9 (midpoint) 14 (distance)
• 10, 15. J (-1,6) K (4,9)
• 10 (midpoint) 15 (distance)

49
Quiz Graph Answers
.D

• Midpoint
• __________
• __________
• __________
• __________
• __________

• Distance
• __________
• __________
• __________
• __________
• __________

.A
.C
.E
.B
50
Slope

• Slope
• The slope of a line is a measure of how tilted
  the line is. A highway sign might say something
  like 6 grade ahead. What does this mean, other
  than that you hope your brakes work? What it
  means is that the ratio of your drop in altitude
  to your horizontal distance is 6, or 6/100. In
  other words, if you move 100 feet forward, you
  will drop 6 feet if you move 200 feet forward,
  you will drop 12 feet, and so on.

51
Slope

• In Geometry, we dont measure percent (usually)
• We measure the rate of change
• We write this as a ratio, with amount of Y
  change on top, and X change on bottom
• Here, the Y coordinate changed 2 and the X
  coordinate changed 4 so the slope is 2/4, or
  1/2

52
Slope

• The next question is, how do we measure the rate
  of change?
• We usually use m to indicate slope, and m
  rise/run, or (Y2-Y1) / (X2-X1)
• Looking at our previous picture, we see how we
  got 2 steps, and 4 steps

We take (4-2)/(5-1) Or 2/4 Or 1/2
53
Slope Intercept Form

• There are three basic equations of a line (also
  known as linear equations)
• The first is Slope Intercept Form
• y mx b
• m the slope of the line, and b the y
  intercept. In other words, if x 0, then where
  does Y intercept the x axis thus Y
  intercept
• NOTE Here you are only given the slope, but can
  easily calculate a point (the Y intercept)

54
Slope Intercept Form

• Y mXb
• You MUST KNOW what m is, and what b is.
• When I ask you on the test What is the slope of
  the line? You need to know to look at the number
  in front of x.
• When I ask you What is the Y intercept? Then
  you need to know to look at b.
• Or, if I ask you If x is zero, what is y? You
  need to know to look at the intercept.

55
Example of Slope Intercept Form

• Graph the line y ¾ X 2
• ALWAYS ASK YOURSELF THIS QUESTION
• IF X IS ZERO, WHAT IS Y?
• Here, if X is 0, Y is 2
• So our first point is (0,2)
• Using slope, we just go up 3 and over 4 (from our
  first point)
• So our second point is (4,5)
• By the way, your calculator is already programmed
  to do this. Just go to the Y button

.
.
56
Standard Form (of a Linear Equation)

• The second linear equation is the Standard
  Form
• The standard form of a Linear Equation is Ax
  By C
• The easiest way to graph using this form is to
  find the Y Intercept, and the X intercept
• In other words, IF X IS ZERO WHAT IS Y, AND
  IF Y IS ZERO WHAT IS X ?this will give you two
  easy points to plot
• NOTE Here, you are not given a point or a slope

57
Example of Standard Form

• Graph 6x 3y 12
• Solve for X 0.
• 3y 12
• Y 4
• So one point is (0,4)
• Solve for Y 0.
• 6x 12
• X 2
• So one point is (2,0)
• Graph the 2 points, and connect the dots.

.
.
58
Example of Standard Form

• Graph -2x 4y -8
• If x is 0
• 4Y -8
• Y -2, Point (0,-2)
• If y is 0
• -2x -8
• X 4, Point (4,0)

.
.
59
Point Slope Form

• The 3rd equation of a line is Point Slope Form
• y- y1 m(x x1)
• NOTE Here you are given a point and the slope

60
Example Point Slope Form

• Write an equation of a line with point (-1,4) and
  slope 3
• Notice, they dont state which equation to use,
  but they give you a point and the slope, so you
  use point/slope
• y-4 3(x (-1))
• y-4 3x 3
• NOTE This is simplified far enough, unless you
  need to graph it. If so, you would simplify one
  step more
• y 3x 7 you could use a calculator now

61
Given 2 Points

• What if you are asked to write an equation of a
  line given 2 points?
• A (-2,3) and B(1,-1)
• 1st, find the slope (y2-y1)/(x2-x1)
• m (-1-3)/(1-(-2)) -4/3
• Now use Point Slope Form
• y- y1 m(x x1)
• y-(3) -4/3(x-(-2)) and simplify
• y-3 -4/3(x 2)

62
Unit 5 Bellringer 1 (10 points)Draw a picture,
write the equation!

• Tom wants to fly a kite higher than Jerry.
• Jerry can fly his kite 50 feet high.
• Tom stands 35 feet away from a point directly
  below his kite.
• If Tom has a kite string that is 65 feet long,
  how high is his kite?
• Is Toms kite higher than Jerrys?

63
Equations for Horizontal Lines

• Write an equation for a horizontal line through
  the point (3,2)
• Notice all points on the line have the same Y
  value. That is, Y 2 no matter what. So a
  horizontal line through the point (3,2) would be
• Y 2 (it doesnt matter what X is)

.
(3,2)
Rule An equation for a horizontal line is Y
the given value of Y
64
Slope of Horizontal Line

• What is the slope of this line?
• Slope
• (y1-y2) / (x1-x2)
• (2 2)/ (-3 3)
• 0/-6
• 0
• Slope of a horizontal line is ZERO!!!

.
.
(3,2)
(-3,2)
65
Equations for Vertical Lines

• Write an equation for a vertical line through the
  point (3,2)
• Notice all points on the line have the same X
  value. That is, X 3 no matter what. So a
  vertical line through the point (3,2) would be
• x 3 (It doesnt matter what Y is)

.
(3,2)
Rule An equation for a vertical line is X the
given value of X
66
Slope of Vertical Line

• What is the slope of this line?
• Slope
• (y1-y2) / (x1-x2)
• (2 - 2)/ (3 3)
• 4/0
• UNDEFINED
• Slope of a Vertical line is UNDEFINED!!!

.
(3,2)
.
(3,-2)
67
Assignment

• Page 194 8-19, 24-37
• Worksheet 3-5

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Unit 5 Quiz 5

• Write the equation for a line in Slope
  Intercept form
• Write the equation for a line in Standard Form
• Write the equation for a line in Point Slope
  form
• Which equation would you use to graph a line if
  you were given the Y intercept and the slope?
  (write the name not the equation)
• Which equation would you use if you were given
  one point and the slope? (write the name, not the
  equation)
• Which equation is easiest to graph if you solve
  for the X and Y intercepts (if x is zero what is
  y, and if y is zero what is x)? (name)
• Using this point (3,2) write an equation for a
  line that is horizontal, and goes through this
  point
• Using this point (3,2) write an equation for a
  line that is vertical, and goes through this
  point
• What is the slope of this line Y 1/2X 5
• Graph this line on your calculator, and leave it
  on so I may check it
• Y 2/3X - 4

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Answers

1. .
2. .
3. .
4. .
5. .

1. .
2. .
3. .
4. .
5. .

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Parallel Lines

• Previously, we learned that if 2 lines are
  parallel, then they do not intersect, and they
  are co-planar. Assuming they are co-planar, how
  do we know that they do not intersect?
• If two lines are parallel, then they have equal
  slopes this means they wont intersect
• If two lines have equal slopes, then they are
  parallel
• By definition Any two (or more) vertical lines
  are parallel
• why do we have to say by definition?
• By definition Any two (or more) horizontal lines
  are parallel
• why do we have to say by definition?

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Checking for Parallel

• How do you check to see if two lines are
  parallel?
• Compare their slopes. If they are equal, then
  they are parallel
• Are lines 4y 12x 20, and y 3x -1 parallel?

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Comparing Lines/Slopes

• Rewrite 4y 12x 20 in slope intercept form
• 4y 12x 20 ? Divide by 4
• y 3x 5
• The other equation was y 3x 1
• Compare slopes
• They have the same slope (it is 3), therefore
  they are parallel
• Also notice that they have different y intercepts
  (5, and -1). If they had the same intercept, they
  would be the same line

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Another Example
.

• Are these lines parallel?
• Compare slopes
• Line a slope
• (4-(-2))/(-2-(-5)) 2
• Line b slope
• (5-(-2))/(4-2) 3 1/2
• The slopes are not equal, the lines are not
  parallel

.
(4,5)
(-2,4)
a
b
.
.
(2,-2)
(-5,-2)
You CANNOT compare by just eyeballing it you
MUST compare slopes
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Writing Equations of Parallel Lines

• Given a line with this equation y - 4x 3
• Given a point at (1,-2)
• Write an equation of a line parallel to the first
  line, going through the given point
• Just write a point slope equation using the slope
  of the first line, and the given point
• y-y1 m(x-x1) or y-(-2) - 4(x-1)
• Which simplifies to y 2 -4(x 1)

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Slopes of Perpendicular Lines
.

• What is the slope of this line?
• The slope is 1(up one over one) or 1/1
• Draw a perpendicular line
• (remember perpendicular means 90 degree angle)
• What is the slope of the new line?
• The slope is -1 (down one over one) or -1/1

.
.
.
.
.
.
.
.
.
.
.
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Compare Slopes
a
c
b
d

• What is the slope of Line a? Of line b?
• Line a Slope 1/2. Line b Slope -2
• What is the slope of line c? Of line d?
• Line c Slope 3. Line d Slope -1/3

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Conclusions

• What can you conclude about the slopes of the
  perpendicular lines in the previous examples?
• If you multiply the slopes together, you always
  get -1. In other words The product of the
  slopes is -1. KNOW THIS!!
• We call this idea negative reciprocals
• 3 x -1/3 -1 or 3/1 x -1/3 -1
• ½ x -2 -1 or ½ x -2/1 -1
• 1 x -1 -1 or 1/1 x -1/1 -1
• In each case, the negative opposite is the
  slope of the second line
• Remember this Flip it, and use the opposite
  sign

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Summary of Perpendicular Lines

• If 2 lines are perpendicular, the product of
  their slopes is -1
• If the slope of 2 lines have a product of -1,
  then the lines are perpendicular
• By Definition Any horizontal line and vertical
  line are perpendicular
• Why by definition?

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Example

• Write an equation for the line perpendicular to
  the line y -3x -5 that goes through the point
  (-3,7)
• Simply use the point slope form
• Identify the negative reciprocal of the slope of
  the first line. The first slope was 3, so the
  negative reciprocal is 1/3
• y-7 1/3(x-(-3)) or y-7 1/3(x3)

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Example

• Write an equation of the line perpendicular to 5y
  x 10 that contains the point (15,-4)
• Solve the first equation for y
• 5y x 10
• y 1/5 x 2
• Negative reciprocal -5. Therefore
• y-(-4) -5(x-15) or y4 -5(x-15)

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Example

• Write an equation of a line perpendicular to y
  2/3x
• Here, we arent given any points, or an
  intercept, so we can use any perpendicular line,
  not one which goes through a specific point. All
  we need is the negative reciprocal
• y -3/2x

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Assignment

• Text Page 201-202 7-21
• Worksheet 3-6

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Unit 5 Quiz 6

• Write the equations for
• Distance Formula
• Point Slope Equation
• Pythagorean Theorem
• Midpoint Equation
• Standard Form Equation
• Slope Intercept Equation
• Slope Equation
• How do you know two lines are parallel on a
  graph?
• How do you know two lines are perpendicular on a
  graph?
• True/False The product of the slopes of parallel
  lines is -1

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Unit 5 Quiz 7

1. What is the first rule of solving radicals?
2. What is the second rule of solving radicals?
3. What is the third rule of solving radicals?
4. What is the first rule of solving exponents?
5. What is the second rule of solving exponents?
6. What is the third rule of solving exponents?
7. What is the fourth rule of solving exponents?
8. What is the rule of the zero exponent?
9. Give an example of an equation for a horizontal
   line
10. Give an example of an equation for a vertical line

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Unit 5 Final Extra Credit (6 points)

• Little Timmy wants to fly his kite higher than
  anybody
• Little Timmy is standing 140 feet away from a
  point on the ground directly under his kite (the
  point is under his kite)
• Little Timmys kite string is 260 feet long
• Little Janie has her kite 220 feet high in the
  air
• How high is little Timmys Kite? (2 pts) Show
  equation
• Who is higher, little Timmy or little Janie? (1
  pt)
• Draw a picture showing little Timmy and his kite.
  Label all 3 sides of the triangle (3 pts)