Mathematics
Grade 8
15 min
Slopes of parallel and perpendicular lines
Slopes of parallel and perpendicular lines
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1
Introduction & Learning Objectives
Learning Objectives
Define parallel and perpendicular lines in terms of their slopes.
Calculate the slope of a line given two points or a linear equation.
Determine if two lines are parallel, perpendicular, or neither by comparing their slopes.
Find the slope of a line that is parallel to a given line.
Find the slope of a line that is perpendicular to a given line.
Apply the concepts of parallel and perpendicular slopes to solve basic geometric problems on a coordinate plane.
Have you ever wondered how architects ensure walls are perfectly straight or how roads can run side-by-side without ever touching? 📐🛣️
In this lesson, we'll explore the special relationships between the slopes of parallel and perpendicular lines. Understanding these relationships is key to analyz...
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Key Concepts & Vocabulary
TermDefinitionExample
SlopeThe measure of the steepness of a line, often described as 'rise over run.' It indicates how much the y-value changes for a given change in the x-value.A line that goes up 3 units for every 2 units it goes right has a slope of 3/2.
Parallel LinesTwo distinct lines in a plane that never intersect, no matter how far they are extended.The opposite sides of a rectangle are parallel lines.
Perpendicular LinesTwo lines that intersect to form a right angle (90 degrees).The adjacent sides of a square are perpendicular lines.
ReciprocalThe number obtained by inverting a fraction (swapping the numerator and denominator). For a number 'a', its reciprocal is '1/a'.The reciprocal of 2/5 is 5/2. The reciprocal of 7 is 1/7.
Negative ReciprocalThe...
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Core Formulas
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Use this formula to calculate the slope ($m$) of a line when you are given two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line.
Slopes of Parallel Lines
If two non-vertical lines are parallel, then their slopes are equal: $m_1 = m_2$.
If you know the slope of one line, you automatically know the slope of any line parallel to it. (Vertical lines are parallel if they have different x-intercepts).
Slopes of Perpendicular Lines
If two non-vertical lines are perpendicular, then the product of their slopes is -1, or one slope is the negative reciprocal of the other: $m_1 \cdot m_2 = -1$ or $m_1 = -\frac{1}{m_2}$.
To find the slope of a line perpendicular to a given line, flip the fraction of the given slope and change its si...
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Challenging
Line L1 passes through (k, 4) and (2, -2). Line L2 is defined by the equation $3x + y = 7$. If L1 is perpendicular to L2, what is the value of k?
A.12
B.20
C.-16
D.4
Challenging
The vertices of a quadrilateral are A(-4, 2), B(0, 5), C(6, -3), and D(2, -6). How can you use slopes to prove that ABCD is a parallelogram?
A.Show that slope of AB = slope of CD, and slope of BC = slope of AD.
B.Show that slope of AB = slope of BC.
C.Show that the product of the slopes of AB and BC is -1.
D.Show that all four sides have the same slope.
Challenging
The vertices of a quadrilateral are P(1, 5), Q(5, 7), R(7, 3), and S(3, 1). By analyzing the slopes of its sides, you can prove this quadrilateral is a rectangle because:
A.Opposite sides are parallel and all four sides have the same length.
B.All four sides have the same slope.
C.Opposite sides are parallel, but adjacent sides are not perpendicular.
D.Opposite sides are parallel and adjacent sides are perpendicular.
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