In the triangle $\mathrm{ABC}$ with vertices $\mathrm{A}(2,3), \mathrm{B}(4,-1)$ and $\mathrm{C}(1,2)$, find the equation and length of altitude from the vertex $\mathrm{A}$.

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Geometry

NCERT Class 11 - Math

Chapter 10

Straight Lines

Section 3

Various Forms of the Equation of a Line

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Video Transcript

you can see that in a triangle abc with tortoises, A. S two comma three Bs for common minus one and C as one comma two. You have to find out the equation as well as the length of the altitude. From the vortex it. So now we're here, as you can see in the figure itself, let abc be the triangle with vortex A, B and C and am over here. Right, This is A M. No, you can see I am is the multitude of the triangle, right? It is the multitude. Now we need to calculate the length as well as a question of the lion am over here. So now, as you can see in altitude, am scoping nuclear too busy. Yeah, So I can just write this in am is perpendicular to busy over here. Okay, now we all know that enable two lines are perpendicular to each other. Then this product of the slopes is always equal to -1. Therefore the slope of am multiplied by slope off we see Maybe equal to -1. All you can say slope of am will be equal to -1 upon slope off Bc Let's say this is a question one. Now, you just need to calculate the slope off BC over here. Okay, so slow off Bc. We know that the line VC is passing through the points full common minus one and one comma two. So whenever you know the two points, you can easily calculate the slope by, I am equal to buy two minus 51 upon X two minus x one. So if you just absolutely values over here, the slope of PC will be 2- of -1 upon Over here. What you'll get 1 -4, so this will be equal to three upon -3. Or you can say -1. Right, so now you know the slope of busy therefore slope of am from equation one, that will be equal to one slope of N is equal to one. Now you just need to find the question of the altitude AM right. So as multitude M passes through the point that is two comma three. And also you know this look that is equal to what. Now the question of a line passing through the point with the slope of M is given by the formula why minus Y one is equal to aim times x minus X one. So you just need to put in the values over. He'll just Plug in X one equal to two while one equal to three and M. S. One. We've already calculated, calculated the slope of the attitude I am over here. So I'm doing that, we'll get y minus three equal to one times x minus two. That means you were here. You will get why minus three equal to x minus two. Or we can say why minus X equal to one. The photo equation of the altitude of am will be why minus X equal to one. Okay, this is the required the equation of the altitude also it was said that besides finding out the equation, you have to find out the length of the contribute as well. So now if we talk about the length so length of AM is equal to the perpendicular distance from point A to busy right is equal to perpendicular. This stands from point A to the line busy. All right now to find a perpendicular distance, we need to find equation of the line busy. So equation of a line Bc over hill will be bought. C stop of line BC was equal to minus one. We've already calculated that slope of the physical to minus one and as you can see it passes through the 10.4 comma minus one. So you can easily find out the equation of the line Bc over here by substituting the values again in the formula y minus y one equal to m times of x minus expert. So on putting up the values will get y plus one equal to minus one times X minus four. Just always will get Y plus X equal to three. Or either you can see X plus y minus three equal to zero. So the question, the line Bc over here is X plus y minus three equal to zero. Now the perpendicular distance D. Of a line X plus B, y plus equal to zero from a point X one, Y one is given by the formula the equal to model is off a X one plus B. Y one plus C upon and the root of a squared plus B squared. So the above the questions of the farm, X plus B, Y plus C. With you have to take A is equal to one, right is equal to one. B is also equal to one and see over here is equal to minus three. So I'm putting up the values you will get B is equal to models of two plus three minus 300 and the root of one plus one, the soldiers, she will get more or less of two upon and the root to all, you can say 200 to upon two which will be able to and the route therefore the length of the altitude length of AM or you can see the length of altitude of the triangle is according to under roof to Yes, so this is the require

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