Understanding parallel and perpendicular lines is important for studying linear equations, especially in Grade 12 Algebra I. These two types of lines have different features that affect how they slope and where they are located on a graph.
What Are They?
Parallel lines are lines in a flat space that never touch each other, no matter how far they go. They always stay the same distance apart.
Slope
The main thing about parallel lines is that they have the same slope. This means they tilt the same way. If you have two lines described by the equations
( y = m_1x + b_1 ) and ( y = m_2x + b_2 ),
they are parallel if ( m_1 = m_2 ).
For example, the lines ( y = 2x + 3 ) and ( y = 2x - 1 ) are parallel because both have a slope of ( 2 ).
Equations
You can write the equations of parallel lines like this:
Graph Appearance
When you draw parallel lines, they look like they are always the same distance from each other. They will never cross, which means there are many possible solutions for lines that aren’t exactly the same.
What Are They?
Perpendicular lines cross each other at a ( 90^\circ ) angle, creating a right angle where they meet.
Slope
Perpendicular lines have a special connection with their slopes. If one line has a slope of ( m_1 ), the slope of the line that is perpendicular to it, ( m_2 ), is the negative version of the fraction of ( m_1 ). This means:
[ m_1 \cdot m_2 = -1. ]
So, if one line has a slope of ( m_1 = 3 ), then the perpendicular line will have a slope of ( m_2 = -\frac{1}{3} ).
Equations
The equations for perpendicular lines can look like this:
Graph Appearance
Perpendicular lines meet at a point and create right angles. When you graph them, they can form squares or rectangles, which helps in understanding shapes in geometry.
In short, the main differences between parallel and perpendicular lines are about how they slope and whether they meet. Parallel lines have the same slope and never cross. Perpendicular lines intersect at right angles, and their slopes are negative versions of each other. Knowing these differences helps with solving linear equations and understanding their graphs in math.
Understanding parallel and perpendicular lines is important for studying linear equations, especially in Grade 12 Algebra I. These two types of lines have different features that affect how they slope and where they are located on a graph.
What Are They?
Parallel lines are lines in a flat space that never touch each other, no matter how far they go. They always stay the same distance apart.
Slope
The main thing about parallel lines is that they have the same slope. This means they tilt the same way. If you have two lines described by the equations
( y = m_1x + b_1 ) and ( y = m_2x + b_2 ),
they are parallel if ( m_1 = m_2 ).
For example, the lines ( y = 2x + 3 ) and ( y = 2x - 1 ) are parallel because both have a slope of ( 2 ).
Equations
You can write the equations of parallel lines like this:
Graph Appearance
When you draw parallel lines, they look like they are always the same distance from each other. They will never cross, which means there are many possible solutions for lines that aren’t exactly the same.
What Are They?
Perpendicular lines cross each other at a ( 90^\circ ) angle, creating a right angle where they meet.
Slope
Perpendicular lines have a special connection with their slopes. If one line has a slope of ( m_1 ), the slope of the line that is perpendicular to it, ( m_2 ), is the negative version of the fraction of ( m_1 ). This means:
[ m_1 \cdot m_2 = -1. ]
So, if one line has a slope of ( m_1 = 3 ), then the perpendicular line will have a slope of ( m_2 = -\frac{1}{3} ).
Equations
The equations for perpendicular lines can look like this:
Graph Appearance
Perpendicular lines meet at a point and create right angles. When you graph them, they can form squares or rectangles, which helps in understanding shapes in geometry.
In short, the main differences between parallel and perpendicular lines are about how they slope and whether they meet. Parallel lines have the same slope and never cross. Perpendicular lines intersect at right angles, and their slopes are negative versions of each other. Knowing these differences helps with solving linear equations and understanding their graphs in math.