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## Geometry (Eureka Math/EngageNY)

### Unit 4: Lesson 2

Topic B: Perpendicular and parallel lines in the Cartesian plane- Parallel & perpendicular lines intro
- Parallel & perpendicular lines from graph
- Parallel & perpendicular lines from graph
- Parallel lines from equation
- Parallel lines from equation (example 2)
- Parallel lines from equation (example 3)
- Perpendicular lines from equation
- Parallel & perpendicular lines from equation
- Writing equations of perpendicular lines
- Writing equations of perpendicular lines (example 2)
- Write equations of parallel & perpendicular lines
- Proof: parallel lines have the same slope
- Proof: perpendicular lines have opposite reciprocal slopes

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# Perpendicular lines from equation

CCSS.Math:

Sal determines which pairs out of a few given linear equations are perpendicular. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What is an equivalant line? It says in the example I have a choise between perpendicular lines, paralel lines, equivalant lines, and so forth. Thank You!(45 votes)
- Equivalent lines are what they say, the same lines! Let's say you're given this example: 5x+y=2 and 20x+4y=8 and you have to figure out what kind of line this is. This would be an equivalent line because you can make the 2 equations the same by multiplying 5x+y=2 by 4 to get the other equation 20x+4y=8 so they are equivalent lines. I really stink at explaining stuff, unlike Sal but I tried my best and I hope this helped! :)(90 votes)

- Could you prove that slopes of perpendicular lines are negative inverses of each other using trigonometry?(15 votes)
- Yes, you can use triangles to prove the slopes are inverses.

Imagine a line from 0,0 to 3,4. At 3,4 you can draw a right triangle with the axis and the point 0,0.

Now imagine a perpendicular line, that intersect at the point 3,4. You can make another triangle from 3,4 to the axis.

The 2 triangles are complementary because the 2 acute angles at the intersection sum 90 degrees, and the 2 triangles have a right angle.

So the relationship between the sides of the 2 triangles is the inverse. And, if the relationship between the sides is the inverse, the slopes of the triangles are inverse also.

But also negative!!! Why? I don't use trigonometry but it's pretty simple.

If a line goes up, positive slope, the perpendicular goes down, negative slope. If a line goes down, negative slope, the perpendicular goes up, positive slope.(16 votes)

- Is there a more rigid definition for perpendicularity? A line of the form y = C has a slope of 0. There is no negative reciprocal for 0, but a line of the form x = C is still perpendicular.(8 votes)
- Thanks for the feedback. The problem with your definition is that it does not cover the case for a horizontal and vertical line which are perpendicular, but the product of their slopes are 0, not -1. I think the wikipedia definition I found covers this limit case.(2 votes)

- At0:30Sal tells me that perpendicular lines must intersect with a right angle.

Parallel lines never meet because they have the same slope...

So what exactly is it when to lines cross, but not with 90 degree angle?(6 votes)- They just intersect. Nothing fancy(3 votes)

- In a square can there be perpendicular line segments?(8 votes)
- All the sides of a square are perpendicular. If you considered the sides as line segments, then yes, there can be perpendicular line segments.(1 vote)

- Do perpendicular lines have to have the same Y intercept? Please help ASAP!(6 votes)
- No, they can have different y-intercepts. To be perpendicular, they only need to have opposite reciprocal slope. For example, the lines, y=3x+8 and y= -(1/3)x-3 would be perpendicular because -1/3 is the opposite reciprocal of 3.(4 votes)

- If you have two equations of lines, how do you tell if they even intersect, not just if they're perpendicular?(3 votes)
- You look at the slope (which either is clearly visible in the equation or you can compute it easily from the coefficients of x and y in the equation). If slopes are the same then the lines are either equivalent (both equations describe the same line) or parallel (and thus do not intersect).

Two lines with different slopes will always intersect.(3 votes)

- How do I know such the equation is or not a Perpendicular line from the equation? What number the slope need be? It isn't the inverse, right?(2 votes)
- For perpendicular lines, the slopes must be opposite reciprocals (different signs and fraction inverted). So -3 and 1/3 or 2/3 and -3/2 are pairs of perpendicular slopes.(4 votes)

- If it doesn't intersect at a 90 degree angle are they still perpendicular? Thanks(1 vote)
- Pupatel,

If intersecting lines are perpendicular they intersect at a 90 degree angle. If lines do not intersect at a 90 degree angle, they are not perpendicular.(4 votes)

- can you give me an example about the perpendicular equation from the graph(2 votes)
- Issam,

A perpendicular line has a slope of the**negative inverse**of the original equations slope.

If y=2x+1 is the first equation, it has a slope of 2.

The negative inverse of 2 is -½ so a perpendicular line would be

y= -½x + ? And value can be used for the ? and the line remains perpendicular.

y= -½x + 3 would be perpendicular to y=2x+1

Here is what the linear equations look like on a graph:

https://www.khanacademy.org/cs/y2x1-and-y-x3/4671527232471040

I hope that helps make it click for you.(2 votes)

## Video transcript

We are asked which of these
lines are perpendicular. And it has to be perpendicular
to one of the other lines, you can't be just perpendicular
by yourself. And perpendicular line, just
so you have a visualization for what for perpendicular lines
look like, two lines are perpendicular if they intersect
at right angles. So if this is one line right
there, a perpendicular line will look like this. A perpendicular line will
intersect it, but it won't just be any intersection,
it will intersect at right angles. So these two lines are
perpendicular. Now, if two lines are
perpendicular, if the slope of this orange line is m-- so let's
say its equation is y is equal to mx plus, let's say
it's b 1, so it's some y-intercept-- then the equation
of this yellow line, its slope is going to be the
negative inverse of this guy. This guy right here is going to
be y is equal to negative 1 over mx plus some other
y-intercept. Or another way to think about
it is if two lines are perpendicular, the product of
their slopes is going to be negative 1. And so you could write that
there. m times negative 1 over m, that's going to be-- these
two guys are going to cancel out-- that's going to be
equal to negative 1. So let's figure out the slopes
of each of these lines and figure out if any of them are
the negative inverse of any of the other ones. So line A, the slope is pretty
easy to figure out, it's already in slope-intercept
form, its slope is 3. So line A has a slope of 3. Line B, it's in standard form,
not too hard to put it in slope-intercept form, so
let's try to do it. So let's do line B over here. Line B, we have x plus 3y
is equal to negative 21. Let's subtract x from both sides
so that it ends up on the right-hand side. So we end up with 3y is equal
to negative x minus 21. And now let's divide both sides
of this equation by 3 and we get y is equal to
negative 1/3 x minus 7. So this character's slope
is negative 1/3. So here m is equal
to negative 1/3. So we already see they
are the negative inverse of each other. You take the inverse of 3, it's
1/3, and then it's the negative of that. Or you take the inverse of
negative 1/3, it's negative 3, and then this is the
negative of that. So these two lines are
definitely perpendicular. Let's see the third
line over here. So line C is 3x plus
y is equal to 10. If we subtract 3x from both
sides, we get y is equal to negative 3x plus 10. So our slope in this
case is negative 3. Now this guy's the negative of
that guy, this guy's slope is a negative, but not the negative
inverse, so it's not perpendicular. And this guy is the inverse of
that guy but not the negative inverse, so this guy is not
perpendicular to either of the other two, but line
A and line B are perpendicular to each other.