A community for students. Sign up today

Here's the question you clicked on:

2bornot2b 3 years ago 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. "

• This Question is Closed
1. 2bornot2b

Here is the solution that I don't understand

2. phi

We start with a given: 2 of the perpendiculars (AD and BE) meet at point O now show that the third perpendicular also meets at point O i.e. if the line segment CO extended to side AB (at point F) is perpendicular to AB then all three perpendiculars meet at O Does that make sense?

3. 2bornot2b

Let me read it just a sec

4. 2bornot2b

This line can you state it in a better way, I don't get it "if the line segment CO extended to side AB (at point F) is perpendicular to AB then all three perpendiculars meet at O "

5. 2bornot2b

OK, I got it... right, you are right, what next?

6. phi

The 3rd perpendicular starts at vertex C. We can draw a line CO (C to O). if we continue, CO intersects the third side AB at F. Now if it turns out that COF is perpendicular to AB, then it is the perpendicular, and it goes through O, the same point as the other 2 perpendiculars

7. 2bornot2b

So we have to show that if the line CO is extended, and at the point where it meets AB it turns out to be perpendicular, then we are done right?

8. phi

yes. And the proof relies on vectors, and the fact that the dot product of two vectors that are perpendicular = 0

9. 2bornot2b

What is the need of that constant l in the equation \(la.(c−b)=0\)

10. phi

good question. It is obviously true, but irrelevant. I would just claim a dot (c-b)= 0

11. phi

Maybe they want to say that AO is too short (i.e. it does not reach the other side), but that we can scale it so that it does.

12. phi

but vectors do not have to "intersect"

13. 2bornot2b

OK

14. phi

I assume that relabeling OA as a, etc makes sense. Do you see how BC= c - b (where all 3 are treated as vectors)?

15. phi

Personally, I always "think" vector addition BC+B= c (using head to tail to add), and then rearrange the vectors to get the difference.

16. phi

*b (not B)

17. 2bornot2b

OK, so I will be going through the solution, and try to understand it, and if I find any problem, I will post it here. Please come again if you find a new post made on this problem :)

18. phi

roger that.

19. 2bornot2b

Thanks a lot

20. 2bornot2b

OK, its clear thank you! I wish I could provide you more medals. It's rare that someone digs in the unanswered questions, like you..

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy