1 SLOPE FORMULA SLOPE FORMULA: PROBLEMS SLOPE: RUN VS RISE. PROBLEMS FALLING OR RISING LINES? SLOPE...

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SLOPE FORMULA

SLOPE FORMULA: PROBLEMS

SLOPE: RUN VS RISE. PROBLEMS

FALLING OR RISING LINES?

SLOPE CASES

PARALLEL VS PERPENDICULAR

PARALLEL VS SKEW AND PERPENDICULAR

Standards 7 and 17

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2

STANDARD 7:

Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and properties of circles.

ESTÁNDAR 7:

Los estudiantes prueban y usan teoremas involucrando las propiedades de líneas paralelas cortadas por una transversal, las propiedades de cuadriláteros, y las propiedades de círculos.

STANDARD 17:

Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.

ESTÁNDAR 17:

Los estudiantes prueban teoremas usando geometría coordenada, incluyendo el punto medio de un segmento, la fórmula de la distancia y varias formas de ecuaciones de líneas y círculos.

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3

Standard 17

m=y1

y2

x1

x2

-

-

Segment AB: A(3,2), B(7,5)y1

y2 x1x2

3 72 5 =

-3-4

= 3 4

Segment CD: C(8,2), D(10,1)y1

y2 x1x2

8 102 1

= 1-2

m =--AB

m =--CD

Slope Formula:

Find the slope for the segments at the right:

1

2

3

4

5

6

7

8

9

21 3 4 5 76 8 9 10 x

y

A

B

CD

The slope is the inclination of a line or segment.

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4

m=y1

y2

x1

x2

-

-

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

y

Using the slope formula below, find the slope for the lines in the coordinate plane:

a

b

c

d

(-8, 7)

(-4, -4)

(-1, 1)

(1, 5)

(3, 2) (10, 2)

(2, -5)

(6, -8)

(1,5) (-1,1)x2

1

x1

-1

y2

5

y1

1

= - 3 4

= 4 2

m =--a

Line a:

= 4

1+1=2

(3,2) (10,2)x2

3

x1

10

y2

2

y1

2 = 0 -7

m =--b

Line b:

=0

(6,-8) (2,-5)x2

6

x1

2

y2

-8

y1

-5=

-3 4

m =--c

Line c:

= -8+5

4

= - 11 4

(-8,7) (-4,-4)x2

-8

x1

-4

y2

7

y1

-4=

11 -4

m =--d

Line d:

= 7+4

-8+4

Standard 17

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m=y1

y2

x1

x2

-

-

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

y

Using the run vs. the rise method, find the slope for the lines in the coordinate plane:

a

b

c

d

= - 3 4

= 4 2

-4-2

m =a

=2

0+7

m =b

=0

+3-4

m =c

= - 11 4

-11

+4m =d

2-

-

4

4-3+

7+

4+

-

11

Line a:

Line b:

Line c:

Line d:

=riserun

+ -+

-

Standard 17

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x

y

c

+-

FALLING TO THE RIGHT

+- +

-+

-

When we move to the right from one point to the other, we go down the right and the slope is negative!

-+

m =c

Slope for c:

= -

Standard 17

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7

Standard 5

x

y

a -

-

- -

m =a

=+

Slope for a:

FALLING TO THE LEFT

-

-

--

-

-

When we move to the left from one point to the other, we go down and the slope is always positive!

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x

y

c

-+

RISING TO THE LEFT

-+-

+-

+

When we move to the left from one point to the other, we go up to the left and the slope is negative!

+-

m =c

Slope for c:

= -

Standard 17

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9

x

y

a

+

++

m =a

=+

Slope for a:

RISING TO THE RIGHT

+

+

+

+

+

+

+

When we move to the right from one point to the other, we go up and the slope is always positive!

Standard 17

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10

x

y

b+

0+

m =b

=0

Slope for b:

Slope of a horizontal line is always 0!

x

ya

-

-

- -

m =a

=+

Slope for a:

Slope of a line that falls to the left is always POSITIVE!

x

y

c-

+

+-

m =c

Slope for c:

= -

The slope of a line that falls to the right is always NEGATIVE!

x

y

+

0m =d

Slope for d:

= not defined!

The slope of a vertical line IS NEVER DEFINED!

SUMMARIZING FINDINGS

d

+

Standard 17

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11

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

y

Parallel Lines have the same slope!

+2+2

m =a

=1

+2+2

m =b

=1

a

b

2+2+

2+2+

PARALLEL VS. PERPENDICULAR

42 6-2-4-6

2

4

6

-2

-4

-6

8 10-8-10

8

-8

10

x

yc

d

2+2+

2-2+

+2-2

m =c

=-1

+2+2

m =d

=1

The slope product of perpendicular lines is -1!

m =d

mc

(-1)(1)

m =a

mb

m =d

mc

-1

PARALLEL PERPENDICULAR

=-1Line a:

Line b:

Line c:

Line d:

Standard 17

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m=y1

y2

x1

x2

-

-

Segment AB: A(2,1), B(6,4)y1

y2 x1x2

2 61 4 =

-3-4

= 3 4

Segment BC: B(6,4), C(9,0)y1

y2 x1

x2

6 94 0

= 4-3

3 4

= 4-3

12-12

= -1

m =--AB

m =--BC

Using the slope formula:

Prove that segments AB and BC are perpendicular:

1

2

3

4

5

6

7

8

9

21 3 4 5 76 8 9 10 x

y

A

B

C

Is the product of the slopes -1?

The product of the slopes is -1. So, They are perpendicular.

Standard 17

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13

Standard 7

PARALLEL VS SKEW

D

A B

C

E F

GH

Lines AB and DC are PARALLEL

AB DC

They are on the same plane and never cross

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Standard 7PARALLEL VS SKEW

D

A B

C

E F

GH

Lines CG and BF are PARALLEL

CG BF

They are on the same plane and never cross

May you name other lines that are parallel? AD HE AB EF

DC HG EH FGPRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15

Standard 7

PARALLEL VS SKEW

D

A B

C

E F

GH

Lines AB and CG are SKEW

They are on different planes and never cross

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16

Standard 7

PARALLEL VS SKEW

D

A B

C

E F

GH

Lines DC and BF are SKEW

They are on different planes and never cross

May you name other lines that are skew? DC and AEHG and AE

FG and AE

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17

Standard 7

D

A B

E F

GH

May you name two planes that are parallel?

Planes ADH and BCG

C

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18

Standard 7

D

A B

E F

GH

May you name two planes that are parallel?

Planes ADH and BCG

C

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19

Standard 7

D

A B

C

E F

GH

May you name two planes that are parallel?

Planes ABC and EFG

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20

Standard 7

D

A B

C

E F

GH

May you name two planes that are parallel?

Planes ABC and EFG

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21

Standard 7

D

A B

C

E F

GH

May you name two planes that are parallel?

Planes ABC and EFG

Also:Planes AEF and DHG

Planes ADH and BCG

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22

Standard 7

PERPENDICULAR LINES

D

A B

C

E F

GH

Lines DC and CG are PERPENDICULAR

They intersect forming a right angle lying on the same plane.

May you name other lines that are perpendicular? DC CB FG CGHG EH DH AD

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23

Standard 7

D

A B

C

E F

GH

May you name two planes that are perpendicular?

Planes DAB and CBF

Also:Planes DAB and DHG

Planes EHG and DAE

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