A community for students.
Here's the question you clicked on:
 0 viewing
vf321
 2 years ago
Can the sum of any two altitudes of a triangle be smaller than one of its legs?
vf321
 2 years ago
Can the sum of any two altitudes of a triangle be smaller than one of its legs?

This Question is Closed

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.0I don't think so, but I'm not sure yet what the easiest proof of that would be.

CliffSedge
 2 years ago
Best ResponseYou've already chosen the best response.0I'm sure the triangle inequality will be in there somewhere, and maybe, since altitudes form right angles, Pythagorean theorem will be useful.

AnimalAin
 2 years ago
Best ResponseYou've already chosen the best response.1Consider an extremely obtuse isosceles triangle. With sides a,a, and b, angles theta (small), theta, and pi  2 theta. The two greater altitudes are b sin theta, so to meet the specification of the problem, 2bsin theta < b implies sin theta less than 1/2. Plenty of angles meet that specification.

vf321
 2 years ago
Best ResponseYou've already chosen the best response.0@AnimalAin That works, thanks. You proved that for triangle ABC, \(a > h_b + h_c\) for some \(a\). What if I asked you to prove that the following: \(b>h_b+h_c\) for some \(b\)?

AnimalAin
 2 years ago
Best ResponseYou've already chosen the best response.1Use the same method, assume b < a/2, and work the inequality similarly. The angles will be smaller, but it can be done.
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.