Prove the given theorem using vectors.
The medians of a triangle meet in a point whose distance from each vertex is two-thirds the length of the median from that vertex.
Prove the given theorem using vectors.
The medians of a triangle meet in a point whose distance from each vertex is two-thirds the length of the median from that vertex.
Hello kullgirl418Continuing from my reply to your previous post, we can generalise the result about the position vector of the mid-point of a line segment, to get the position vector of a point dividing a line segment in a given ratio, and it's this:
The position vector of the point that divides the line segment in the ratio is given by:So suppose the triangle is is the mid-point of and divides in the ratio . Then:
, as before using the mid-point formulaand:
, using the more general result aboveNow you'll see that is the point that's two-thirds of the way down the median . I'll leave it to you to show that it's also two-thirds of the way down the other medians as well. (Although this is pretty obvious since the expression for is symmetrical in and .)
Grandad
Hello kullgirl418 is the mid-point of the side of the triangle. So if the position vectors of and are and , we use the 'mid-point formula' that I showed you in your first posting to get .
It's not terribly helpful to try to imagine what the vector looks like, but if you want to, you'll need to mark a point somewhere - it doesn't matter where - to represent the origin. Then join to - that's the vector , or for short.
Grandad
You may refer to http://www.mathhelpforum.com/math-he...oincident.html
Hello kullgirl418It's clear that you're not really understanding vectors at all, so read this carefully.
I didn't label B and C as vectors - I referred to their position vectors:...in other words, their positions in relation to some arbitrary origin - usually denoted by O.
If you're not clear what that phrase 'their positions in relation to some arbitrary origin' means, let me unpack it a little - but you'll have to do the work. So:
- Draw a triangle ABC - don't make it a special triangle; try not to make it right-angled or isosceles.
- Mark M as the mid-point of BC.
- Somewhere else on the paper mark a point O. It's easiest if it's outside the triangle and doesn't lie on any of its sides when they're produced.
- Join O to A, B, C and M.
Then the position vectors of the points A, B, C and M in relation to O are the displacement vectors and . (A displacement vector is a description of how you move from one point to another. So the displacement vector describes - using whatever language you'd like - how you would move from O to A.)
To make it simpler to write, we describe these position vectors - these 'movement descriptions' - using lower-case letters:The triangle law of addition is then simply two movement descriptions' combined into one. So when we write:we're just saying:
So when we use lower-case letters this becomes:
in other wordswhich I described in an earlier reply to you as a very important result.
Finally, let's show you again where the 'mid-point' formula comes from.
Writing this important result again, but using the points B and M this time:But the movement from B to M is just one-half of the movement from B to C, since M is the mid-point of BC. So
Knowing that (there's that result again), we can say:I hope that helps to make things clearer.
Grandad