Mathematics
Grade 7
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 slope as a measure of steepness.
Calculate the slope of a line from a graph or two given points.
State the relationship between the slopes of parallel lines.
State the relationship between the slopes of perpendicular lines.
Determine if two given lines are parallel, perpendicular, or neither by comparing their slopes.
Ever wonder why train tracks never meet, or how builders make sure walls stand perfectly straight? 🛤️🏠
In this lesson, you'll discover how the 'steepness' of lines, called slope, helps us understand special relationships between lines – specifically, when they are parallel or perpendicular. This knowledge is key to understanding shapes and structures around us.
Real-World Applications
Architecture and construction (...
2
Key Concepts & Vocabulary
TermDefinitionExample
SlopeSlope is a measure of the steepness and direction of a line. It tells us how much a line rises or falls for a given horizontal distance. It's often described as 'rise over run'.A ramp that goes up 2 feet for every 10 feet horizontally has a slope of 2/10 or 1/5.
RiseThe 'rise' refers to the vertical change between two points on a line. It's the change in the y-coordinates.If a line goes from a y-coordinate of 2 to a y-coordinate of 5, the rise is 5 - 2 = 3.
RunThe 'run' refers to the horizontal change between two points on a line. It's the change in the x-coordinates.If a line goes from an x-coordinate of 1 to an x-coordinate of 4, the run is 4 - 1 = 3.
Parallel LinesParallel lines are lines in a plane that are always...
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Core Formulas
Slope Formula
$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$
Use this formula to calculate the slope ($m$) of a line given two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. The 'rise' is the change in y, and the 'run' is the change in x.
Slopes of Parallel Lines
If two non-vertical lines are parallel, then their slopes are equal: $m_1 = m_2$.
To check if two lines are parallel, calculate their slopes. If the slopes are the same, the lines are parallel. (Note: Vertical lines are parallel to each other but have undefined slopes).
Slopes of Perpendicular Lines
If two non-vertical lines are perpendicular, then the product of their slopes is -1: $m_1 \cdot m_2 = -1$. This also means one slope is the negative reciprocal of the othe...
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Challenging
A triangle has vertices at points A(1, 1), B(3, 5), and C(7, 3). Is this a right-angled triangle?
A.Yes, because sides AB and BC are perpendicular.
B.Yes, because sides AB and AC are perpendicular.
C.Yes, because sides AC and BC are perpendicular.
D.No, it is not a right-angled triangle.
Challenging
Line A passes through (0, 1) and (2, 5). Line B passes through (2, 5) and a point (x, 4). For what value of x will Line A be perpendicular to Line B?
A.0
B.3
C.4
D.1
Challenging
The slope of Line 1 is a/b. Line 2 is perpendicular to Line 1. If the 'run' (horizontal change) of Line 2 is 'a', what is its 'rise' (vertical change)?
A.b
B.-a
C.a
D.-b
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