Video: Finding the Equation of a Line Perpendicular to a Given Line

A straight line 𝐿 has the equation 𝑦 = 3𝑥 − 2. Find the equation of the line perpendicular to 𝐿 that passes through the point (4, 4).

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Video Transcript

A straight line 𝐿 has the equation 𝑦 equals three 𝑥 minus two. Find the equation of the line perpendicular to 𝐿 that passes through the point four, four.

Okay, when we try to solve this problem and we want to find the equation of this new line, the keyword is perpendicular, as this gives us an idea of the relationship between our straight line 𝐿 and this new line. The way this helps us is because actually it talks about the slopes between the two lines. Because if they’re perpendicular, it means the product to the slopes is equal to negative one, but how that can help us find out the slope of our new line?

If we look at another definition of the relationship of the two lines, we can see that the slope of perpendicular lines are the negative reciprocal of one another. And actually, we can use this trying to work out what the slope will be of our new line. Well, looking at our original equation, we’ve got 𝑦 equals three 𝑥 minus two. Well, we know that the coefficient of 𝑥 is going to be the slope of 𝐿. So that means we can say the slope of 𝐿 is three and we know that because of 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the slope and 𝑐 is the 𝑦-intercept.

Great! So we now got the slope of 𝐿. We’re gonna find the slope of our new line and we’re gonna do that using our definition for the relationships between the slopes of perpendicular lines and the original line.

First of all, we know it’s negative because it’s saying it’s the negative reciprocal of one another. So in this case, because it was positive three is the slope of 𝐿, then the slope of 𝑝, our perpendicular, is going to be negative.

And then the second part is gonna be one-third. And the reason it’s one-third is because of this word here reciprocal. And reciprocal what it’s actually means is what you multiply a number by to get one. For example, three multiplied by third gives you three-thirds, which equals to one.

Great! And we can check that we’ve got it right as well by using the first definition we looked at: the product of the slopes needs to be equal to negative one. Well, three multiplied by negative a third is equal to negative one. So yes, great! We’ve now found the slope of our new line.

We can now move on to work out the equation of the line, and we’re gonna do that using this. And this is the point-slope equation, where 𝑚 is the slope, which we discussed before, and 𝑎 and 𝑏 are the 𝑥- and 𝑦-coordinates of a point that is on that line. So we can now start to sub these values in. So 𝑎, so we got 𝑦 minus 𝑎. 𝑎 is the 𝑦-coordinate; well, we’ve got this point over here, which is four, four. So our 𝑦-coordinate is four.

So we have 𝑦 minus four is equal to our slope, which is negative a third multiplied by and then inside the parentheses, we have 𝑥 minus 𝑏; well, 𝑏 is our 𝑥-coordinate, which is also four, which gives us 𝑦 minus four is equal to negative a third 𝑥 minus four.

Great! So now all we need to do is to simplify this to have our final equation. Okay, to simplify it to the next stage, we’ll expand the parentheses to let 𝑦 minus four is equal to negative a third 𝑥. And then be careful of this bit; we’ve got negative a third multiplied by negative four, which will give us positive four over three.

Fab! We’re almost there, just one more stage. And to do that, we need to add four to both sides of the equation. To enable us to do that more easily, I’d actually convert this four into thirds, so that would give us 12 thirds. So now, we can calculate our final equation, which is 𝑦 is equal to negative a third 𝑥 plus 16 over three; it’s 16 over three as you got four over three plus 12 over three gives us 16 over three. Fantastic! So we’ve now managed to find the equation of the line perpendicular to 𝐿 that passes through the point four, four.

So just a quick recap to see how we’ve done that. So first of all, if you find an equation of another line, check to see if it’s parallel. If it’s parallel, it will have the same slope. If it’s perpendicular, it will have the negative reciprocal of the slope. Once you’ve got that, you use the point-slope equation and substitute the value of a point that you have, and then you’ll be able to find your new equation.

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