unit 1 parallel & perpendicular lines. slope-intercept review section 1

UNIT 1

Parallel & Perpendicular Lines

Slope-Intercept Review

Section 1

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

Example

y = 2x – 1 Slope: m = 2 =

y-intercept: b = -1

21

Example Set 1

Identify the slope and y-intercept of each line, given its equation: y = ½x – 3 y = 3x y = -5 – 2x

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

Identifying Parallel/Perpendicular Lines

Section 2

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

Example Set 3

Find the opposite reciprocal of the following numbers:

3 -

-4

1

31

13

13

25

Parallel, Perpendicular, or Neither?

Parallel

Neither

Neither

Perpendicular

Neither

Perpendicular

Finding the Equation of a Parallel/Perpendicular Line through a Point

Section 3

Announcements

EXTENSION: Homework Packet due Monday

Unit 1 Test POSTPONED until Monday

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?

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

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?

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!

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!

Wrap Up

Exit Slip Remember, homework packet and test

now for MONDAY the 27th

Midpoint Formula

Section 4

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

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

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)

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?

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).

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.

Wrap Up

Exit Slip Remember, homework packet and test

now for MONDAY the 27th

Distance Formula

Section 5

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?

Finding the distance continued Horizontal

distance = 8 Vertical

distance = 6 Pythagorean

Theorem:

a2 + b2 = c2

62 + 82 = 100 d = √100 = 10

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

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?

Wrap Up

Exit Slip Remember, homework packet and test

now for MONDAY the 27th

Finding the Perpendicular Bisector

Section 6

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

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

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

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