3.43.4 Find and Use Slopes of LinesBell Thinger

1. Evaluate if a = 5, b = 2, c = 1, and d = 7.a – b c – d

ANSWER 12

–

2. Solve .x – 33 – 4 = 1

5

ANSWER 145

ANSWER 32

3. What is the reciprocal of ?23

3.4

3.4Example 1Find the slope of line a and line d.

SOLUTION

Slope of line a: m = – 1

Slope of line d: m = 4 – 0 6 – 6 = 4

0

which is undefined.

y2 – y1 x2 – x1

= = 4 – 2 6 – 8 = 2

– 2

y2 – y1 x2 – x1

=

3.4Guided PracticeFind the slope of the line.

1. Line b 2ANSWER

2. Line c 0ANSWER

3.4

3.4Example 2

Find the slope of each line. Which lines are parallel?

SOLUTION

Find the slope of k1 through (– 2, 4) and (– 3, 0).

= – 4 – 1 = 4

m21 – 5 3 – 4

= = 4

Find the slope of k2 through (4, 5) and (1, 3).

= – 4 – 1

m1 = 0 – 4– 3 – (– 2 )

3.4Example 2

m3– 2 – 3 5 – 6

= = – 5 – 1

= 5

Find the slope of k3 through (6, 3) and (5, – 2).

Compare the slopes. Because k1 and k2 have the same slope, they are parallel. The slope of k3 is different, so k3 is not parallel to the other lines.

3.4Guided Practice

3. Line m passes through (–1, 3) and (4, 1). Line t passes through (–2, –1) and (3, – 3). Are the two lines parallel? Explain how you know.

Yes; they have the same slope. ANSWER

3.4Example 3

SOLUTION

Line h passes through (3, 0) and (7, 6). Graph the line perpendicular to h that passes through the point (2, 5).

= 64 = 3

2m1 = 6 – 0

7 – 3

STEP 1Find the slope m1 of line h through (3, 0) and (7, 6).

3.4

STEP 2

Find the slope m2 of a line perpendicular to h. Use the fact that the product of the slopes of two perpendicular lines is –1.

m2 = – 2 3

23

Multiply each side by .

STEP 3Use the rise and run to graph the line.

Example 3

m2 =32

– 1 Slopes of perpendicular lines

3.4 Example 4

Quadrilateral ABCD has vertices A(7, 3), B(5, 6), C(–1, 2), and D(1, –1). Find the slope of the sides and the length of the sides. Then tell whether quadrilateral ABCD is a rectangle, a square, or neither. Explain your reasoning.

SOLUTIONFind slopes.

slope of AB

slope of BC

slope of CD

slope of DA

6 35 7

32

2 61 5

46

23

1 21 ( 1)

32

3 ( 1)7 1

46 2

3

3.4

Find lengths.2 2(5 7) (6 3) AB 13

2 2( 1 5) (2 6) 52

13

52

2 2[1 ( 1)] ( 1 2)

2 2(7 1) [3 ( 1)]

BC

CD

DA

Quadrilateral ABCD is a rectangle, but it is not a square.

The products of the slopes of consecutive sides is

, so ABCD has four right angles. The four sides

are not all the same length.

3 2 12 3

Example 4

3.4Guided Practice

4. Line n passes through (0, 2) and (6, 5). Line m passes through (2, 4) and (4, 0). Is n m? Explain.

ANSWER Yes; the product of their slopes is – 1.

3.4Example 5Roller Coasters

During the climb on the Magnum XL-200 roller coaster, you move 41 feet upward for every 80 feet you move horizontally. At the crest of the hill, you have moved 400 feet forward.

a. Making a Table: Make a table showing the height of the Magnum at every 80 feet it moves horizontally. How high is the roller coaster at the top of its climb? SOLUTION

a.

The Magnum XL-200 is 205 feet high at the top of its climb.

3.4Example 5

b. Calculating : Write a fraction that represents the height the Magnum climbs for each foot it moves horizontally. What does the numerator represent?

c. Using a Graph: Another roller coaster, the Millennium Force, climbs at a slope of 1. At its crest, the horizontal distance from the starting point is 310 feet. Compare this climb to that of the Magnum. Which climb is steeper?

3.4Example 5SOLUTION

a.

The Magnum XL-200 is 205 feet high at the top of its climb.

Slope of the Magnumb. = 41 8080 80 = 0.5125

1

The numerator, 0.5125, represents the slope in decimal form.

= riserun

4180=

3.4Example 5Use a graph to compare the climbs. Let x be the horizontal distance and let y be the height. Because the slope of the Millennium Force is 1, the rise is equal to the run. So the highest point must be at (310, 310).

c.

The graph shows that the Millennium Force has a steeper climb, because the slope of its line is greater (1 > 0.5125).

ANSWER

3.4Guided Practice

Line qANSWER

Line q passes through the points (0, 0) and (– 4, 5). Line t passes through the points (0, 0) and (–10, 7). Which line is steeper, q or t ?

5.

3.4Exit Slip1. Find the slope of the line containing the points

(4, – 3) and (5, 2).

5ANSWER

2. Line k passes through the points (– 1, 2) and (3, 5).Line n passes through the points (3, 7) and (6, 3).Are lines k and n parallel, perpendicular, or neither?

PerpendicularANSWER

3. A highway has a grade of 7 percent. For each 200feet it goes horizontally, how many feet does it rise?

14 ftANSWER

3.4Homework

Pg 181-184#12, 14, 15, 24, 32