3.3 slopes of lines. objectives find slopes of lines use slope to identify parallel and...
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3.3 Slopes of Lines3.3 Slopes of Lines
ObjectivesObjectives
Find slopes of lines
Use slope to identify parallel and perpendicular lines
What is Slope?
The slope (m) of a line is the number of units the line rises or falls for each unit of horizontal change from left to right.
In other words, slope is the ratio of vertical rise or fall to its horizontal run.
m = riserise
runrun
yy22 - - mm = =
xx22 - - xx11
yy22 -- yy11
xx22 -- xx11
In algebra, we also learned a formula for slope called theIn algebra, we also learned a formula for slope called the slope formula.
Subtraction order is the sameSubtraction order is the same
CORRECT
xx11 -- xx22
yy22 -- yy11 Subtraction order is differentSubtraction order is different
INCORRECT
The Slope FormulaThe Slope Formula
Remember to keep your subtraction of Remember to keep your subtraction of the coordinates in the proper order.the coordinates in the proper order.
yy11
From (–3, 7) to (–1, –1), go down 8 units and right 2 units.
Find the slope of the line.
Answer: – 4
Example 1a:Example 1a:
Use the slope formula.
Answer: undefined
Find the slope of the line.
Let be and be .
Example 1b:Example 1b:
Find the slope of the line.
Answer:
Example 1c:Example 1c:
Find the slope of the line.
Answer: 0
Example 1d:Example 1d:
Slopes of Slopes of ║ and ║ and Lines Lines
Finally, recall from algebra that lines which are║ or have mathematical relationships.
║ lines have the same slope.
i.e. If line l has a slope of ¾ and line m is ║to line l then it also has a slope of ¾.
lines have opposite reciprocal slopes.
i.e. If line a has a slope of 2 and line b is to line a then it has a slope of – ½.
Slope PostulatesSlope Postulates
Postulate 3.2Two non-vertical lines have the same slope if they are ║.
Postulate 3.3Two non-vertical lines are if the
product of their slopes is -1.
Determine whether and are parallel, perpendicular, or neither.
Example 3a:Example 3a:
The slopes are not the same,
The product of the slopes is
are neither parallel nor perpendicular.
Answer:
Example 3a:Example 3a:
Answer: The slopes are the same, so
Determine whether and are parallel, perpendicular, or neither.
Example 3b:Example 3b:
are | | .
Answer: perpendicular
Answer: neither
a.
b.
Determine whether and are parallel, perpendicular, or neither.
Your Turn:Your Turn:
Graph the line that contains Q(5, 1) and is parallel to with M(–2, 4) and N(2, 1).
Substitution
Simplify.
Slope formula
Example 4:Example 4:
The slopes of two parallel lines are the same.
Graph the line. Answer:
The slope of the line parallel to
Start at (5, 1). Move up 3 units and then move left 4 units.
Label the point R.
Example 4:Example 4:
Graph the line that contains R(2, –1) and is parallel to with O(1, 6) and P(–3, 1).
Answer:
Your Turn:Your Turn:
Assignment
Pg 142-143 #16-32 (evens)