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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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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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
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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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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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