ccss content standards g.gpe.5 prove the slope criteria for parallel and perpendicular lines and use...

Content Standards

G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Mathematical Practices

4 Model with mathematics.

7 Look for and make use of structure.

8 Look for and express regularity in repeated reasoning.

You used the properties of parallel lines to determine congruent angles.

• Find slopes of lines.

• Use slope to identify parallel and perpendicular lines.

• slope

• rate of change

Find the Slope of a Line

A. Find the slope of the line.

Substitute (–3, 7) for (x1, y1) and (–1, –1) for (x2, y2).

Answer: –4

Slope formula

Substitution

Simplify.

Find the Slope of a Line

B. Find the slope of the line.

Substitute (0, 4) for (x1, y1) and

(0, –3) for (x2, y2).

Answer: The slope is undefined.

Slope formula

Substitution

Simplify.

Find the Slope of a Line

C. Find the slope of the line.

Answer:

Slope formula

Substitution

Simplify.

Substitute (–2, –5) for (x1, y1) and (6, 2) for (x2, y2).

Find the Slope of a Line

D. Find the slope of the line.

Answer: 0

Slope formula

Substitution

Simplify.

Substitute (–2, –1) for (x1, y1) and (6, –1) for (x2, y2).

A.

B.

C.

D.

A. Find the slope of the line.

A. 0

B. undefined

C. 7

D.

B. Find the slope of the line.

A. 0

B. undefined

C. 3

D.

D. Find the slope of the line.

Use Slope as Rate of Change

RECREATION In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the annual sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2015?

Understand

Use the data given to graph the line that models the annual sales y as a function of the years x since 2000. The sales increase is constant. Plot the points (0, 48.9) and (5, 85.9) and draw a line through them.

You want to find the sales in 2015.

Use Slope as Rate of Change

Plan

Find the slope of the line. Use this rate of change to find the amount of sales in 2015.

Solve

Use the slope formula to find the slope of the line.

The sales increased at an average of $7.4 million per year.

• 2005 to 2015 is 10 years• 7.4(10) = $74 million more than in 2005

• 89.5 + 74 = $159.9 million will be the total sales in 2015.

Use Slope as Rate of Change

Answer: Thus, the sales in 2015 will be about $159.9 million.

Check From the graph we can estimate that in 2015, the sales will be a little more than $150 million. Since 159.9 is close to 150, our answer is reasonable.

A. about 251.5 million

B. about 166.3 million

C. about 180.5 million

D. about 194.7 million

CELLULAR TELEPHONES Between 1994 and 2000, the number of cellular telephone subscribers increased by an average rate of 14.2 million per year. In 2000, the total subscribers were 109.5 million. If the number of subscribers increases at the same rate, how many subscribers will there be in 2010?

• Assignment p. 192-193 1, 2, 21, 25, 27

Determine Line Relationships

Step 1 Find the slopes of and .

Determine whether and are parallel, perpendicular, or neither for F(1, –3), G(–2, –1), H(5, 0), and J(6, 3). Graph each line to verify your answer.

Determine Line Relationships

Step 2 Determine the relationship, if any, between the

lines.The slopes are not the same, so and are not parallel. The product of the slopes is

So, and are not

perpendicular.

Determine Line Relationships

Answer: The lines are neither parallel nor perpendicular.

Check When graphed, you can see that the lines are

not parallel and do not intersect in right angles.

Determine whether AB and CD are parallel, perpendicular, or neither for A(–2, –1), B(4, 5), C(6, 1), and D(9, –2)

Use Slope to Graph a Line

First, find the slope of .

Slope formula

Substitution

Simplify.

Graph the line that contains Q(5, 1) and is parallel to MN with M(–2, 4) and N(2, 1).

Use Slope to Graph a Line

The slope of the line parallel to through Q(5, 1) is .

The slopes of two parallel lines are the same.

Graph the line.

Draw .

Start at (5, 1). Move up 3 units and then move left 4 units.

Label the point R.

Answer:

Graph the line that contains R(2, –1) and is parallelto OP with O(1, 6) and P(–3, 1).

A. B.

C. D. none of these

What is the graph of the line that has slope 0 and contains the point (–3, –4)?

A. B.

C. D.

• Assignment p. 193 29, 31, 35, 41