2bornot2b
  • 2bornot2b
I am trying to solve the following problem, and I have a solution which I don't understand, can you help me? "Show that in a triangle the perpendiculars drawn from the vertices are concurrent. "
Mathematics
chestercat
  • chestercat
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2bornot2b
  • 2bornot2b
Here is the solution that I don't understand
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Mani_Jha
  • Mani_Jha
Exactly which part u dont understand? It is all about triangle law of vector addition and dot products
2bornot2b
  • 2bornot2b
What has the solution shown so that its clear that the perpendiculars are concurrent.

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Mani_Jha
  • Mani_Jha
Any two lines have to intersect(unless they are parallel). Now, if a third line passes through the intersection of these two lines, then the three are concurrent. In this solution, they have taken two perpendiculars. Through their intersection point a line has been drawn, and if that line happens to be the altitude through C, then all perpendiculars have to be concurrent. That's what we have to prove here. The dot product of two perpendicular lines is 0. So, we must prove that the dot product of AB and CF is zero. I hope it helped, if not please say
2bornot2b
  • 2bornot2b
Just a sec, let me read it and understand
2bornot2b
  • 2bornot2b
Have you seen they have used a constant like \(l\) while writing \[la(c-b)=0\] Whats the need of that \(l\)
Mani_Jha
  • Mani_Jha
Well, let me guess. Here OA=a. But actually we see that AD is perpendicular to BC and also intersects it. So, they took the length of AD instead of AO. So, AD/AO=l is assumed. so that al is the length of AD. But I dont think there is any compulsion of taking l and m here.
2bornot2b
  • 2bornot2b
Do you have any better solution for it?
Mani_Jha
  • Mani_Jha
Of course there is a geometrical solution to this. Well, let me work on it. I will post it as soon as i complete it
2bornot2b
  • 2bornot2b
No thank you, I need to make it through vector
2bornot2b
  • 2bornot2b
Thats mandatory for me
2bornot2b
  • 2bornot2b
OK, thank you!
2bornot2b
  • 2bornot2b
OK so mani jha provided with the explanation that " The dot product of two perpendicular lines is 0. So, we must prove that the dot product of AB and CF is zero. I hope it helped, if not please say", but it didn't quite satisfy me. Can you think of any better explanation. Does this solution have anything to do with vector equation?

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