Triangle altitude equation
About this video
Sal finds the equation of the line containing the altitude of triangle ABC from A, perpendicular to base BC. He finds the slope of BC (-11/2), takes the negative reciprocal to get the altitude's slope (2/11), and uses point-slope form with A(-2, 0) to get y = 2/11(x + 2), or y = 2/11 x + 4/11.
You'll be able to
- find the slope of a segment from the coordinates of its endpoints.
- find the slope of a perpendicular line using the negative reciprocal.
- write the equation of an altitude of a triangle using point-slope form.
Key words
- altitude
- A segment from a vertex of a triangle that meets the opposite side (the base) at a right angle.
- perpendicular
- Meeting at a right angle (90 degrees).
- negative reciprocal
- Flip a fraction and change its sign; perpendicular lines have slopes that are negative reciprocals, like -11/2 and 2/11.
- slope
- Change in y divided by change in x between two points on a line.
- point-slope form
- The equation y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Transcript
- [Narrator] We're told figure triangle ABC is a triangle. I guess that's a little bit redundant. And they draw it right over here for us. Write the equation of the line that contains the altitude of triangle ABC, that is perpendicular to base BC, and passes through the point A.
So pause this video, have a go at that before we do this together. All right, first let's try to visualize what they're even talking about. So they want an altitude, but an altitude, there's at least three altitudes, depending on how you want to count them that we could draw here, but they're saying perpendicular to base BC.
So this is base BC here. So we want an altitude perpendicular to that that goes through point A. So it's going to go through this point. It's going to be perpendicular through that side. So this is the altitude. We want the equation of this line that contains this altitude.
So I guess really we're finding the equation of this entire line here and it contains this altitude. So how do we do that? Well, the first step is, let's see if we can figure out the slope of that line. How do we do that? Well, we have a clue. That line is going to be perpendicular to base BC.
So if we can figure out the slope of BC, then we could use that to figure out the slope of this line here. So let's figure out the slope of BC. So this one right over here, I could write that M is going to be equal to change in Y over change in X. So change in Y, negative eight minus three, over negative three, minus negative five.
So this is going to be equal to, in our numerator we have negative 11. And in our denominator, see this is negative three, plus positive five, if you subtract a negative, so this is going to be positive 2 So the slope is negative 11 halves for this base right over here.
So what is going to be the slope for the line in question? Well, the slope of that altitude, and we could call this the slope of the base if we want for B or BC, if we, let's write it as BC for segment BV. I know that's a little bit messy, but what's going to be the slope of this one?
Well, it is perpendicular and a perpendicular line is gonna have the negative reciprocal. So this line right over here, its slope is going to be not just the reciprocal, which is 2 over 11, actually it would be the negative, it would be negative 2 over 11, but we're gonna take the negative of that.
So it's positive 2 over 11. So instead of negative 11 halves, we're gonna take the negative. So that'd make it positive. And then we flip it, we take the reciprocal. So this slope right over here is going to be 2 elevenths. So we know the slope and we know a point that sits on that line, point A.
In fact, we knew it had to go through point A. So how do we figure out the equation of that line? Well, we have a point and we have a slope. Maybe point slope form will be helpful. Point slope form is Y minus Y1 is going to be equal to our slope times X minus X1 and Y1 and X1 are these coordinates, X and Y coordinates.
So when X is equal to negative 2, when X is negative 2, Y is equal to zero. So this one just goes away and our slope is 2 elevenths. So we get the left side simplifies to Y is equal to, we get 2 over 11 times x minus negative 2, X plus 2. So this over here, we could say we're done.
This is an equation for that line. We could try to express it in other forms, say MX plus B form. We could distribute this 2 elevenths. So we could also write it like this. Y is equal to 2 elevenths X plus 4 elevenths. That's another way you do it. You could also manipulate it, get into standard form, but that's, they're not asking us to do all of those different types of things.
Now the last thing I'll point out is why is this even useful? Well, what's really useful about this is you could do this with any triangle, where they're giving you these three points. And if you know the equation of this altitude, then you could figure out also this intersection point.
You could figure out the length of that altitude and then you could use that to figure out the area of this triangle. You could use this to figure out the area of any triangle if you have the three coordinate points. We haven't gotten that far, but this is the first step to go on that journey.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.