Mathematics
Grade 10
15 min
Parallel, perpendicular, intersecting
Parallel, perpendicular, intersecting
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1
Introduction & Learning Objectives
Learning Objectives
Identify the relationship between two lines (parallel, perpendicular, or intersecting) given their equations in any form.
Determine the slope of a line that is parallel to a given line.
Determine the slope of a line that is perpendicular to a given line.
Write the equation of a line that is parallel to a given line and passes through a specific point.
Write the equation of a line that is perpendicular to a given line and passes through a specific point.
Use slope criteria to prove whether two lines are parallel or perpendicular.
Algebraically determine the point of intersection for two non-parallel lines.
Ever noticed how the streets on a city map can run alongside each other, cross at perfect corners, or meet at an angle? 🗺️ Let's explore the alge...
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Key Concepts & Vocabulary
TermDefinitionExample
Parallel LinesTwo distinct lines in a coordinate plane that never intersect. They have the same steepness.The lines y = 2x + 3 and y = 2x - 1 are parallel because they both have a slope of 2.
Perpendicular LinesTwo lines in a coordinate plane that intersect to form a right (90°) angle.The line y = 2x + 1 is perpendicular to the line y = -1/2x + 5.
Intersecting LinesLines that cross at exactly one point. This category includes perpendicular lines, but more generally refers to any two lines that are not parallel.The lines y = 3x + 2 and y = -x + 6 are intersecting because their slopes (3 and -1) are different.
SlopeA measure of the steepness and direction of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two poi...
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Core Formulas
Rule for Parallel Lines
Two non-vertical lines are parallel if and only if their slopes are equal. `m_1 = m_2`
Use this rule to check if two lines are parallel or to find the slope needed for a new parallel line. The y-intercepts must be different for the lines to be distinct.
Rule for Perpendicular Lines
Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. `m_1 * m_2 = -1` or `m_1 = -1 / m_2`
Use this rule to check if two lines form a right angle or to find the slope needed for a new perpendicular line.
Rule for Intersecting Lines
Two lines intersect if and only if their slopes are not equal. `m_1 ≠ m_2`
This is the general case for any two lines that cross. If their slopes are not equal, they must intersect at...
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Challenging
For what value of 'k' will the line kx - 6y = 10 be parallel to the line y = -2x + 5?
A.3
B.-12
C.12
D.-3
Challenging
Determine the value of 'k' such that the line passing through (k, 8) and (1, 5) is perpendicular to the line y = (1/2)x + 9.
A.-0.5
B.2.5
C.1
D.-1.5
Challenging
A line segment has endpoints A(-3, 1) and B(5, 7). What is the equation of the perpendicular bisector of this segment?
A.y = (3/4)x + 19/4
B.y = (-3/4)x + 13/4
C.y = (4/3)x + 8/3
D.y = (-4/3)x + 16/3
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