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.
CCSS 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.
Dec 18, 2015
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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
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