UNIT 1 Parallel & Perpendicular Lines. Slope-Intercept Review Section 1.
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Transcript of UNIT 1 Parallel & Perpendicular Lines. Slope-Intercept Review Section 1.
![Page 1: UNIT 1 Parallel & Perpendicular Lines. Slope-Intercept Review Section 1.](https://reader036.fdocuments.net/reader036/viewer/2022062309/5697bfe81a28abf838cb673f/html5/thumbnails/1.jpg)
UNIT 1
Parallel & Perpendicular Lines
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Slope-Intercept Review
Section 1
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Slope-Intercept Form of a Line
y = mx + b
x and y are variables m and b are numbers m is the slope
“Rise over run” b is the y-intercept
Point where the line crosses the y-axis
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Example
y = 2x – 1 Slope: m = 2 =
y-intercept: b = -1
21
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Example Set 1
Identify the slope and y-intercept of each line, given its equation: y = ½x – 3 y = 3x y = -5 – 2x
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Example 2
Given equation 2x + 3y = 7, can we immediately find the slope and y-intercept?
No! We must first put it in slope-intercept form!
Get the y by itself:2x + 3y = 7
-2x -2x 3y = -2x + 7 3
y = -2/3x + 7/3
Slope: -2/3 y-intercept: 7/3
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Identifying Parallel/Perpendicular Lines
Section 2
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Parallel vs. Perpendicular
Parallel: Two lines in the same plane that never intersect Parallel Lines have equal slopes
Perpendicular: Two lines that intersect to form a 90° angle Perpendicular lines have
opposite reciprocal slopes Opposite: different signs (+/-) Reciprocal: flip the fraction
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Example Set 3
Find the opposite reciprocal of the following numbers:
3 -
-4
1
31
13
13
25
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Parallel, Perpendicular, or Neither?
Parallel
Neither
Neither
Perpendicular
Neither
Perpendicular
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Finding the Equation of a Parallel/Perpendicular Line through a Point
Section 3
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Announcements
EXTENSION: Homework Packet due Monday
Unit 1 Test POSTPONED until Monday
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Finding the Equation of a Parallel Line
Sample Problem: Write the equation of a line parallel to the
line y = -2x + 3 that passes through the point (2,1).
Think back: What do you know about the slopes of parallel lines?
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Finding the Equation… continuedWrite the equation of a line parallel to the line
y = -2x + 3 that passes through the point (2,1).
1. Find the slope m of a parallel line.
2. Plug slope into y = mx + b.
3. Plug x and y-values into equation from step 2.
4. Simplify, solve for b.
5. Rewrite equation using new m and b values.
m = -2
y = -2x + b
1 = -2(2) + b 1 = -4 + b +4 +45 = b
y = -2x + 5
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Finding the Equation of a Perpendicular Line
PREDICT: How might the steps be different if we’re finding the equation of a perpendicular line through a point?
HINT: What do we know about the slopes of perpendicular lines?
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Finding the Equation… continued
Write the equation of a line perpendicular to the line
y = -2x + 3 that passes through the point (2,1).
1. Find the slope m of a perpendicular line.
2. Plug slope into y = mx + b.
3. Plug x and y-values into equation from step 2.
4. Simplify, solve for b.
5. Rewrite equation using new m and b values.
m = y = x + b
1 = (2) + b 1 = 1 + b -1_ -1____ 0 = b
y = x
121
2
12
12
Opposite reciprocal!
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Practice
Find the equation of a line parallel to the line-3x + y – 2 = 4 that passes through the point (-2,-4). y = 3x + 2
Find the equation of a line perpendicular to the line -x – 2y = 6 that passes through the point (4,-1). y = -2x + 7
I will pick people to come up to the board for each problem!
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Wrap Up
Exit Slip Remember, homework packet and test
now for MONDAY the 27th
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Midpoint Formula
Section 4
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Midpoint
What is the midpoint of a line? Point on the line equidistant from the two
endpoin
Midpoint Formula:
Notice it’s just the average of the two x-values and the average of the two y-values!
2,
22121 yyxx
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Example
What is the midpoint of the line segment with endpoints at A(3, -4) and B(5, -1)?
214
,2
53
25
,28
25
,4
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Practice
Find the midpoint of line segment AB with endpoints A(4, -6) and B(-4, 2).
Find the midpoint of line segment CD with endpoints C(0, -8) and D(3, 0).
Find the midpoint of line segment XY with endpoints X(-3, -7) and Y(-1, 1)
Find the midpoint of line segment LN with endpoints L(12, -7) and N(-5, -2)
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Finding the Other Endpoint
How do we find the other endpoint if we know the midpoint and first endpoint? Example: Find the endpoint B of line
segment AB, with endpoint A(0,-5) and midpoint M(2,-3).
Try coming up with the answer by graphing the endpoint and the midpoint. How many spaces up and to the right
should the other endpoint be?
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Math Challenge
Can you come up with a way to find the other endpoint algebraically (without graphing)? Example: Find the endpoint B of line
segment AB, with endpoint A(0,-5) and midpoint M(2,-3).
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Practice
M is the midpoint of QR with Q(-3, 5) and M(7, -9). Find the coordinates of R.
D is the midpoint of CE with E(-3, -2) and D(5, 1). Find the coordinates of C.
M is the midpoint of LN with L(0, 0) and M(-2, -8). Find the coordinates of N.
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Wrap Up
Exit Slip Remember, homework packet and test
now for MONDAY the 27th
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Distance Formula
Section 5
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How do we find the distance between two points?
Example: Line segment AB has endpoints A(5, 4) and B(3,-2). Find the length of AB.
Hint: Can you figure it out by graphing AB?
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Finding the distance continued Horizontal
distance = 8 Vertical
distance = 6 Pythagorean
Theorem:
a2 + b2 = c2
62 + 82 = 100 d = √100 = 10
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Distance Formula
We can also plug A(-3, -2) and B(5, 4) into this formula:
Example:
2 2(5 3) (4 2)d
2 22 1 2 1( ) ( )d x x y y
2 2(8) (6) 64 36 100 10d
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Practice
The endpoints of RT are R(-1,-2) and T(5, 6). What is the length of RT?
The endpoints of AB are A(0, 7) and B(-3, 11). Find the length of AB.
Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards, as shown in the diagram below. What is the length of the diagonal, in yards, that Tanya runs?
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Wrap Up
Exit Slip Remember, homework packet and test
now for MONDAY the 27th
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Finding the Perpendicular Bisector
Section 6
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What does it mean to bisect something?
Bisect: to split in half PREDICT: What is a perpendicular
bisector? Line that is perpendicular to a line segment
and splits it in half
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Finding the Equation of a Perpendicular Bisector from Two Endpoints Example: Find the equation of the
perpendicular bisector of the line segment with endpoints A(2, 3) and B(-2, -5).
Similar to Monday’s lesson with finding the equation of a perpendicular line, with two differences: You have to calculate the slope using the
slope equation You must calculate the midpoint and plug it in
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Equation of a Perpendicular Bisector continued
Find the equation of the perpendicular bisector of the line segment with endpoints
A(2, 3) and B(-2, -5).
1.Calculate the slope using the slope formula.
2. Find the opposite reciprocal.
3.Plug it into the equation y = mx + b.
12
12
xxyy
m
2235
48
2
21
m
bxy 21
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Equation of a Perpendicular Bisector continued
A(2,3) and B(-2,-5)4.Find the midpoint of AB.
5.Plug coordinates of midpoint into equation.
6.Solve for b.7.Rewrite Equation with m and b.
253
,2
22
22
,20 1,0
b )0(21
1
b 01 1b
121
xy