Parallel and Perpendicular Lines

Parallel lines

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.

	Equation for red line	Slope	Equation for blue parallel line	Slope

Look at the slopes of the two lines in the second row of the table. Are they the same or are they different?

What about the slopes of the two lines in the third row? Are they the same or are they different?

What is the slope of any line that is parallel to the line $y=3x-2$?

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 $y_1=4$.

Use the sliders to find a point-slope equation for a line that passes through the point $(1,3)$ and is parallel to the line $w=-x+3$ (the red line).

Can you find an equation for a different line that passes through the point $(1,3)$ and is parallel to the line $w=-x+3$?

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.

Point	Equation	$m$	$x_1$	$y_1$
$(-3,2)$			$-3$
$(2,4)$				$4$

Perpendicular lines

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 $w=2x+2$.

Use the slider for $m$ to make the graph of $y=mx+b$ perpendicular to the red line. What is the equation you found? (Leave $b$ at the value 0.)

What is the blue line’s slope (as a fraction)?

What happens when you leave $m$ at the value you found and slide $b$? Does the blue line stay perpendicular to the red line?

Can you find a line perpendicular to $w=2x+2$ that has a positive slope?

Use the sliders to find an equation for another line that is perpendicular to $w=2x+2$.

What is its slope (as a fraction)?

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.

	slope of red line	slope of blue line

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 $-1$.

What do you get if you multiply the slope of the red line in the second row of the table by the slope of the blue line in the second row of the table?

What do you get if you multiply the slope of the red line in the third row of the table by the slope of the blue line in the third row of the table?

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 $$-1/m$$.

If the red line had slope 4, what would be the slope of the perpendicular blue line?

If the red line had slope $$-1/6$$, what would be the slope of the perpendicular blue line?

What is the slope of any line that is perpendicular to the line $y=3x-2$?

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.

As you change $m$, do the lines remain perpendicular?

As you slide $m$ from 1 to 2, does the red line become steeper or less steep?

As you slide $m$ from 1 to 2, does the blue line become steeper or less steep?

Can you find any value of $m$ which makes the slopes of both the red line and the blue line positive?

Use the sliders to find a point-slope equation for a line $y-y_1=m(x-x_1)$ that passes through the point $(3,2)$ and is perpendicular to the line $w=-2x+3$ (the red line).

Can you find an equation for a different line that passes through the point $(3,2)$ and is perpendicular to $w=-2x+3$?

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.

point	equation	slope
$(-3,2)$
$(2,4)$

Do the slopes follow the $$-1/m$$ rule given above?

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 $c$.

Use the sliders to find equations for four lines that go through the points $\{(0, 2), (1, 4), (2, 1), (3, 3)\}$ plotted on the grid to the left and have a pattern similar to the one shown above.

$y=$ $w=$ $s=$ $u=$

Which pairs of lines are parallel?

$y$ and are parallel. and are parallel.

Which pairs of lines are perpendicular?

$y$ and $w$ are perpendicular. $y$ and are perpendicular. $w$ and are perpendicular. and are perpendicular.

Change the value of $m$ and notice how it affects the shape of the graph. Do the parallel and perpendicular relationships you found change when you change $m$?