Lesson 3-3 Slopes of Lines
Standardized Test Practice: Transparency 3-3 5-Minute Check on Lesson 3-2 In the figure, m4 = 146. Find the measure of the following angles. 2 7 10 4. 11 5. Find m11 + m6 6. In the figure to the right what is the measure of ABC? 146 34 146 34 180 Standardized Test Practice: A A 77 B 84 C 89 D 91
Objectives Find slopes of lines Use slopes to identify parallel and perpendicular lines
Vocabulary Slope – symbol m; the vertical rise divided by the horizontal run (or m = y/x) A horizontal line has a slope of zero (no change in y; y = 0) A vertical line has an undefined slope (no change in x; x = 0) Rate of change – describes how a quantity changes over time
Postulates 3.2 – Two non-vertical lines have the same slope if, and only if, they are parallel. 3.3 – Two non-vertical lines are perpendicular if, and only if, the product of their slopes is -1.
Review of Slopes Slope, m, is a rate of change (∆y / ∆x) between two points (x1, y1) and (x2, y2) y x y2 – y1 ∆y m = --------- = ------ x2 – x1 ∆x Two lines are parallel (never intersect), if they have the same slope (myellow = 1) Two lines are perpendicular (forms a right angle), if their slopes multiply together to get -1 (mpink = -1)
Find the slope of the line. From (–3, 7) to (–1, –1), go down 8 units and right 2 units. Answer: – 4
Find the slope of the line. Use the slope formula. Let be and be . Answer: undefined
Find the slope of the line. Answer:
Find the slope of the line. Answer: 0
a. Find the slope of the line. Answer:
b. Find the slope of the line. Answer: 0
c. Find the slope of the line. Answer: 2
d. Find the slope of the line. Answer: undefined
RECREATION For one manufacturer of camping equipment, between 1990 and 2000, annual sales increased by $7.4 million per year. In 2000, the total sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2010? Slope formula
Simplify. Multiply each side by 10. Add 85.9 to each side. The coordinates of the point representing the sales for 2010 are (2010, 159.9). Answer: The total sales in 2010 will be about $159.9 million.
Determine whether and are parallel, perpendicular, or neither. The slopes are not the same, The product of the slopes is are neither parallel nor perpendicular. Answer:
Determine whether and are parallel, perpendicular, or neither. Answer: The slopes are the same, so are parallel.
a. b. Determine whether and are parallel, perpendicular, or neither. Answer: perpendicular Answer: neither
Summary & Homework Summary: The slope of the line is the ratio of its vertical rise to its horizontal run Parallel lines have the same slope Perpendicular lines have slopes whose product is -1 (also known as negative inverses) Homework: pg 142-143: 15, 16, 19, 20, 25, 27, 29, 30