Main content

### Course: High school geometry > Unit 6

Lesson 4: Parallel & perpendicular lines on the coordinate plane# Parallel & perpendicular lines from graph

The slopes of parallel lines are equal, and the slopes of perpendicular lines are opposite reciprocals. This is a worked example of determining whether given lines are parallel or perpendicular.

## Want to join the conversation?

- At4:10Sal says slope but writes M. Why?(14 votes)
- That's a good question.

I did some internet research. I found that according to http://mathworld.wolfram.com/Slope.html , it is not known for sure why slope is called m but perhaps it's because "climb" in English translates to "montar" in French.

Have a blessed, wonderful day!(29 votes)

- Why does sal subtract -3-0 when the slope formula is y2-y1/ x2-x1 .

Shouldn’t it be 0-(-3)(12 votes)- It's fine either way, wether you start from x1,y1 or x2,y2. Just think of them as a ratio of height to the base to determine the "slope".(4 votes)

- is math related to science?(4 votes)
- Sometimes, Math is separated into theoretical and practical, much of what you learn in school math classes is theoretical, but word problems often attempt to move from theoretical to practical. Much of science is related to the practical aspects of math, but not all of science is math related.(5 votes)

- At6:55there is a fraction over a fraction i always get confused with these.Is there any video available in khan academy or an article for explanation?(3 votes)
- This whole topic of negative inverses of slopes, is this explained in more detail on Khan Academy?(3 votes)
- Could you have lines that are both parallel and perpendicular ?(3 votes)
- no because perpendicular lines always intersect .parallel lines never intersect(1 vote)

- why is it so that the multiplication of the gradients of both lines which are perpendicular to each other is equivalent to -1? How do you prove this?(1 vote)
- Check out the following lesson here on Khan Academy https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-parallel-perpendicular-eq/v/proof-that-perpendicular-lines-have-negative-reciprocal-slope

It shows the proof that perpendicular lines have a negative reciprocal slope. :)(1 vote)

- in the 2nd one why do I divide -12/8 by 4?(2 votes)
- the reason he did that is he simplified the fraction, to get a better example of what the fraction is to decimal form.(1 vote)

- Is there a formula to find where two lines intersect based on their equations?(1 vote)
- It's not a formula, but rather a method.

It is called solving a system of equations.

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/v/trolls-tolls-and-systems-of-equations

I'm not going to explain it since KA covered it already, but I will give a general example for solving intersection.

y = m₁x + c₁

y = m₂x + c₂

Since for an intersection, both equations will have the same y-value (As it's the same point), hence

m₁x + c₁ = m₂x + c₂

x * (m₁ - m₂) = c₂ - c₁

x = (c₂ - c₁) / (m₁ - m₂)

And substitute this x back to any one of the equation and you get your intersection.(2 votes)

- It means there is no solution between two equations if two lines are parallel, but are there any benefits in knowing two lines are perpendicular?(1 vote)
- Yes, because by knowing 2 lines are perpendicular, you know the angle between them is a right angle (90 degree). This is really important for a lot of geometry problems.

For example, you have 3 lines forming a triangle. By knowing 2 of them are perpendicular to each other, you will know that this triangle is a right triangle. Thus you can use trigonometry or Pythagorean's Theorem on this triangle.

Another example is a theorem stating that if you have a circle and an inscribed triangle inside this circle, if this triangle is right triangle, then the opposite side of this right angle is the diameter. Now checking perpendicular becomes handy if you want to find the area of the circle etc.

Also, if you have an isosceles triangle ABC where AB = AC. Now suppose you have a line from A onto BC, and you would like to know that is this line the perpendicular bisector of this triangle. Since property of isosceles triangle states that if this line is perpendicular to BC, then this is the altitude and the bisector of BC, you can once again use your knowledge on finding perpendicular lines.(2 votes)

## Video transcript

- [Instructor] In this video,
we're gonna do a couple of examples that deal with
parallel and perpendicular lines. So you have parallel, you have perpendicular, and of course you have lines
that are neither parallel nor perpendicular. And just as a bit of a review, if you've never seen this before, parallel lines they never intersect. So let me draw some axes. So if those are my
coordinate axes right there, it's my X-axis, that is my Y-axis, if this is a line that
I'm drawing in magenta, a parallel line might
look something like this. It's not the exact same line, but they have the exact same slope. If this moves a certain amount, if this change in Y over
change in X is a certain amount this change change in Y over
change in X is the same amount. And that's why they never intersect. So they have the same slope. Parallel lines have the same slope. Perpendicular lines, depending
on how you wanna view it, they're kind of the opposite. Let's say this is some line,
a line that is perpendicular to that, will not only intersect the line, it won't only intersect the line, it will intersect it at a right angle. At a 90 degree angle. At a 90 degree angle, and
I'm not going to prove it for you here, I actually prove it in the linear algebra playlist, but a perpendicular lined slope, let's say this one right here,
let's say that yellow line has a slope of M, then this orange line that's perpendicular to the yellow line is going to have a slope
of negative one over M. Their slopes are gonna
be the negative inverse of each other. Now, given this information, let's look at a bunch
of lines and figure out if they're parallel, if
they're perpendicular, or if they are neither. And to do that, we just have
to keep looking at the slopes. So let's see, they say
one line passes through the points four and negative three and negative eight, zero. Another line passes through
the points negative one, negative one, and negative two, six. So let's figure out the
slopes in each of these lines. I'll first do this one in pink. So this slope right here, so line one. So I'll call it slope one. Slope one is, let's just say it is, well I'll take this as
the finishing point, so negative three minus
zero, remember change in Y, negative three minus zero, over four, minus negative eight. Four minus negative eight, so this is equal to negative three, over, this is the same
thing as four plus eight, negative three over 12, which
is equal to negative 1/4. Divide the numerator by
the denominator by three, that's this line, the first line. Now what about the second line? The second line, the slope
for that second line is, well let's take, here negative one, negative one minus six, over negative one, minus negative two is equal
to negative one minus six is negative seven, over
negative one minus negative two, that's the same thing as
negative one plus two, well that's just one. So the slope here is negative seven. So here their slopes are neither equal, so they're not parallel. Nor are they the negative
inverse of each other, so this is neither, so
this is neither parallel or perpendicular. Neither parallel nor perpendicular, so these two lines they intersect but they're not going to
intersect at a 90 degree angle. Let's do a couple more of these. So I have here, once
again, one line passing through these points, and
then another line passing through these points. So let's just look at their slopes. So this one in green, what's the slope? The slope of the green one,
I'll call that the first line. We could say, let's see change in Y, so we could do negative two minus 14, over, I did negative two
first, so I'll do one first, over one minus negative three. So negative two minus 14 is negative 16. One minus negative three
is the same thing as one plus three, that's over four, so this is negative four. Now what's the slope of that
second line right there? So we have the slope of that second line, let's say five minus, so we'll say five minus negative three, that's our change in Y, over negative two minus zero. So this is equal to five
minus negative three that's the same thing as five plus three, that's eight. And then negative two
minus zero is negative two, so this is also equal to negative four. So these two lines are parallel. These two lines are parallel. They have the exact same slope. And I encourage you to find the equations of both of these lines and
graph both of these lines and verify for yourself that
they are indeed parallel. Let's do this one. Once again, this is just an
exercise in finding slopes. So this first line has those points. Let's figure out it's slope. The slope of this first line. One line passes through these points, so let's say three minus negative three, that's our change in Y, over three minus negative six. So this is the same thing as
three plus three which is six over three plus six which is nine. So the first line has a slope of 2/3. What is the second line's slope? This is the second line there. That's the other line
passing through these points. So the other line's slope, let's see, we could say negative eight minus four, over two minus negative six. So what is this equal to? Negative eight minus four is negative 12. Two minus negative six,
that's the same thing as two plus six, right? The negatives cancel out. So it's negative 12 over
eight, which is the same thing if we divide the numerator
and the denominator by four, that's negative 3/2. Notice these guys are the
negative inverse of each other. If I take negative one over 2/3, that is equal to negative one times 3/2, which is equal to negative 3/2. These guys are the negative
inverses of each other. You swap the numerator and
denominator make 'em negative, and they become equal to each other, so these two lines are perpendicular. I encourage you to find the equations, I've already got the slopes for you, but find the equations
of both of these lines plot them and verify for yourself that they are perpendicular.