Parallel and Perpendicular Lines

In this lab 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.

Parallel lines

Question 1. 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
	$w=2x+1$	2
	$w=-2x-4$
	$w=x+4$

What do you notice about the slopes of the pairs of parallel lines?

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.

Question 2. Click on to see the example for question 2. The blue line is the graph of $y-y_1=m(x-x_1$), which is the 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 parallel to $w=-x+3$ and that 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.

Question 3. Click on to see the example for question 3. 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? Why or why not?

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

What is its slope (as a fraction)?

Question 4. Click on each Next button in the table below to see the example for that row of the table. Each graph will show a pair of perpendicular lines. Enter the slope of each blue line in the table as a fraction.

	slope of red line	slope of blue line
	−1/3
	1/2
	5

Question 5. What do you notice about the slopes of each of the pairs of lines in question 4?

If two lines are perpendicular and the slope of one of them is $m$, then the slope of the other line is $-1/m$.

Question 6. What is the slope of any line that is perpendicular to the line $y=3x-2$?

Question 7. Click on to see the example for question 7. The slope of the red line is $m$ and the slope of the blue line is $-1/m$. Use the slider to change the value of $m$. Describe what happens. Do the lines remain perpendicular?

Question 8. Click on to see the example for question 8. 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. Use the sliders to check your answers.

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

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

Question 9. Click on to see the example for question 9. The graphs of $y$ and $s$ have slope $m$, while the graphs of $w$ and $u$ have slope $-1/m$.

Graph

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

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

Which pairs of lines are parallel?

Which pairs of lines 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 $m$?

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