SlopeSection 4.6

ByJessica McMinn, Lizzy Shields, and

Taylor Ross

What is Slope?

• Definition: Slope is the steepness of a non vertical line. It is measured by the following formula:

• Slope is the difference in the y coordinates over the difference in the x coordinates.

• Slope = rise over run

Slope Formula

• A was located (2,2)• B was located at (-3,-4)• Plug the numbers into the formula. Slope (m) formula: 2- (-4) = 6 2- (-3) 5

A (2,2)

B (-3,-4)

Types of Slope

• Positive = Rising

• Negative = Falling

• Zero = Horizontal

• No Slope = Vertical

Theorems Involving Slope

• Theorem 26: If two nonvertical lines are parallel then their slopes are equal

• Theorem 27: If two slopes of two nonvertical lines are equal then the lines are parallel

• Parallel lines => same slope

• Same slopes => parallel lines

Theorems Involving Slope (cont’d)• Theorem 28: If two lines

are perpendicular and neither is vertical, each line’s slope is the opposite reciprocal of the other’s

• Theorem 29: If a line’s slope is the opposite reciprocal of another line’s slope the two lines are perpendicular

• Nonvertical perpendicular lines <=> slopes of these lines are opposite reciprocals

• Slopes of these lines are opposite reciprocals <=> nonvertical perpendicular lines

Sample Problems

Problem 1:If F = (2,8) and Z = (-4, 6), find the slope of FZ.

Solution:Use the slope formula:

m=Y2-Y1

X2-X1

= 8-6 2-(-4)

= 2 6

Sample Problems

• Is Triangle ABC a right triangle if A is (-3,2), B (-1,5), and • C (5,1)?• First find the slopes of all the sides

• m AB =

• m AC =

• m BC = • Because the slopes of AB and BC are opposite reciprocals, angle B is a

right angle. Therefore, triangle ABC is a right triangle.

2

3

)3(

2

1

5

8

1

8

1

5

1

3

2

3

2

6

4

5

1

1

5

A

B

C

Sample Problems

A (3,2)

B (-1, 8)

C (-5,2)

D

Given: triangle ABCa.) Find the slope of median CD.

Use the midpoint formula:

The midpoint of BA (D) =

m CD =

b.) Find the slope of altitude AE

m BC =

Because slope of the altitude is the opposite reciprocal of the slope BC, m AE = -2/3.

5

1

52/10

12/2

2

3

8

1

4

3

5

2

1

5

D =(1,5)

E

2

3

4

6

5

2

1

8

Practice Problems

Find the slope of the line determined by each pair of points:

1. ( 2, 3) and ( 5, 6) 1

2. (-4, 2) and (1, 5) 3/5

3. (3, 0) and (-2, 4) -4/5

4. (8, 3) and (-6, 7) -2/7

5. (3, 5) and (8, -3) -8/5

6. (0, 4) and (3, 7) 1

Practice Problems

A (-3,3)B (3, 3)

D (-4,-2) C (2, -2)

Are AB and DC parallel?

Yes. They both have zero slope. When two lines have the same slope, the two lines are parallel.

Practice ProblemsIf angle A’BA is rotated 90 degrees about the origin in a clockwise direction, what will be the new coordinates of point A’?

(2,1)

Works Cited• "4 Types of Slopes." Algebra.com. Algebra.com, Web. 14

Jan 2010. <http://www.algebra.com/cgi-bin/plot- formula.mpl?expression=graph+%28600%2C600%2C- 20%2C20%2C-20%2C+20%2C+-2x+%2B+4%2C+-2x+%2B+13%29&x=0003>.

• "Cartesian Plane." Fym.edu. Web. 17 Jan. 2010. <http://fym.la.asu.edu/~tturner/MAT_117_online/Cartesian_plane /right_triangle_example.jpg>.

• "Coordinate Plane." Google.com. Google. Web. <http://images.google.com/images?gbv=2&hl=en&sa=1&q=coor dinate+plane+&aq=f&oq=&aqi=&start=0>

Works Cited• "Math Images." Ritter.com. Web. 17 Jan. 2010.

<http://ritter.tea.state.tx.us/student.assessment/resources/online/2006/grade7/math/images/1graphicaa.gif>.

• "Parallel Line Conjunctures." Geom.uiuc. Web. 17 Jan. 2010. <http://www.geom.uiuc.edu/~dwiggins/conj16.html>.

• Kuca, . "Mrs. Kuca's Home Page." http://dk044.k12.sd.us/algebra1.htm. Web. 14 Jan 2010.

<http://dk044.k12.sd.us/algebra1.htm>.

• Learningwave Online, . "4 Types of Slopes." www.learningwave.com. 1999. LearningWave Online, a division of HRM Video, Web. 14 Jan 2010. http://www.learningwave.com/lwonline/algebra_section2/slope2.html

Extra Credit Video(Double Click the Picture)