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In this lesson you will study parallel and perpendicular lines and notice relationships between their slopes. You will also learn how to find equations for lines that are parallel or perpendicular to each other and that go through specific points.
The red line on the grid to the left is the graph of $w=2x+1$. Use the slider to change the value of $m$ in the equation $y=mx-3$, and find an equation for a line parallel to the line $w=2x+1$. Complete the first row of the table below. (The equations for the lines are shown below the grid.) Then click on each Next button and repeat this process to complete each row.
If two lines are parallel, they have the same slope.
This blue line is the graph of $y-y_1=m(x-x_1)$, which is a point-slope equation for a line. If you want to graph a line with slope 2 that passes through the point $(3,4)$, you should set $m=2$, $x_1=3$ and $y_1=4$.
For each point in this table, use the sliders to find an equation for a line which is parallel to $w=-x+3$ and goes through the given point.
Perpendicular lines are lines that form 90° angles. So far we have looked at parallel lines and seen that two parallel lines have the same slope. In this section we will look at perpendicular lines and find out if there is any relationship between their slopes.
The red and green lines are perpendicular. The equation for the red line is $w=2x+2$.
For each row in the table below, the graph will show a pair of perpendicular lines. Enter the slope of each blue line in the table.
Look at the first row of the table in perpLines. If you multiply the slope of the red line ($$-1/3$$) by the slope of the blue line (3), you get $-1$.
As you noticed in perpMult, multiplying the slopes of any two perpendicular lines gives $-1$. Another way to say this is:
If two lines are perpendicular and the slope of one of them is $m$, then the slope of the other line is $$-1/m$$.
The slope of this red line is $m$ and the slope of the blue line is $$-1/m$$. Use the slider to change the value of $m$ and answer the following questions.
For each point in this table, find an equation for the line through the point that is perpendicular to the line $w=-2x+3$. Write your equation in point-slope form. Enter the slopes in fraction form (using a ‘/’). Use the sliders to check your answers.
There are four equations graphed on the grid to the left. The graphs of $y$ and $s$ have slope $m$, while the graphs of $w$ and $u$ have slope $$-1/m$$. The graphs of $y$ and $w$ have $y$-intercept 2, while the $y$-intercept of $s$ is $b$ and the $y$-intercept of $u$ is $c$.