Mathematics
Grade 10
15 min
Parallel, perpendicular, and intersecting lines
Parallel, perpendicular, and intersecting lines
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define and differentiate between parallel, perpendicular, and intersecting lines.
Determine the relationship between two lines given their equations in slope-intercept or standard form.
Calculate the slope of a line that is parallel or 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.
Apply the properties of parallel and perpendicular lines to solve problems involving geometric figures on a coordinate plane.
Ever wonder how city planners design perfectly straight, non-colliding roads or how architects ensure corners are perfect right angles? 🏙️ It all comes down to the math...
2
Key Concepts & Vocabulary
TermDefinitionExample
Intersecting LinesTwo or more distinct lines in a plane that cross each other at exactly one point. They have different slopes.The lines y = 2x + 1 and y = -x + 4 are intersecting lines. They cross at the point (1, 3).
Parallel LinesTwo distinct lines in a plane that never intersect. They have the same slope but different y-intercepts.The lines y = 3x + 2 and y = 3x - 5 are parallel. They both have a slope of 3.
Perpendicular LinesTwo lines that intersect to form a right angle (90 degrees). Their slopes are negative reciprocals of each other.The lines y = 2x + 1 and y = -1/2x + 3 are perpendicular. Their slopes, 2 and -1/2, are negative reciprocals.
SlopeA measure of the steepness of a line, calculated as the ratio of the vertical change (rise) to the horizontal chan...
3
Core Formulas
Parallel Lines Slope Condition
For two non-vertical lines, `l_1` and `l_2`, with slopes `m_1` and `m_2` respectively, the lines are parallel if and only if `m_1 = m_2`.
Use this rule to determine if two lines are parallel or to find the slope of a line that needs to be parallel to another. This applies only if the y-intercepts are different; otherwise, the lines are coincident.
Perpendicular Lines Slope Condition
For two non-vertical lines, `l_1` and `l_2`, with slopes `m_1` and `m_2` respectively, the lines are perpendicular if and only if `m_1 * m_2 = -1`, or `m_2 = -1/m_1`.
Use this rule to check if two lines form a right angle. The slopes must be negative reciprocals of each other. This rule does not apply to horizontal and vertical lines, which are always perpendicular....
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
For what value of k are the lines (k-1)x + 2y = 3 and 4x + (k+1)y = 7 parallel?
A.k = 0
B.k = 3 or k = -3
C.k = 4
D.k = -3
Challenging
Find the equation of the perpendicular bisector of the line segment with endpoints A(-3, 2) and B(5, 6).
A.y = -2x + 6
B.y = 1/2x + 7/2
C.y = -2x + 2
D.y = -1/2x + 9/2
Challenging
Line L₁ has the equation y = mx + c, where m ≠ 0. Line L₂ is perpendicular to L₁ and passes through the origin (0,0). What is the x-coordinate of the intersection point of L₁ and L₂ in terms of m and c?
A.c / (m² + 1)
B.-mc / (m² + 1)
C.mc / (m² - 1)
D.-c / m
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free