June 14th, 2018, 02:06 AM  #1 
Member Joined: Aug 2017 From: India Posts: 54 Thanks: 2  Maximum area of triangle
Sorry for lot of questions. But please help me to solve this problem My Attempt: Area = (1/2) * Base * Height; Area = (1/2) * p*q; Maximum is infinite value because p and q can take any value. Am I correct? Please help. I think I am nowhere near the correct thinking. Last edited by skipjack; June 14th, 2018 at 02:23 AM. 
June 14th, 2018, 02:36 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2207 
Given the values of p and q, you might as well fix the position of the line segment AB, as doing so won't affect the required area. AB can be called the base of the triangle. The distance of C from AB can be called the height of the triangle. The area of the triangle is maximized when this distance is maximized, because the area is half the product of this distance and p, the length of the base. To maximize the distance of C from AB, BC must be perpendicular to AB, and the maximized distance is then q, the length of BC. The area of triangle ABC is then pq/2.

June 14th, 2018, 03:12 PM  #3 
Senior Member Joined: Jun 2014 From: USA Posts: 528 Thanks: 43 
From the textbook History of Mathematical Thought (Kline, I believe), it actually hasnt been proven the sum of the angles of a triangle must always add up to 180 degrees. It had something to do with parallel lines not having to be exacty parallel before being provably nonintersecting. The sum of the angles of a triangle can be more than 180 degrees given the assumption. This from a class I took about 15 years ago so sorry not more specific. I assume the above would impact the area. Hope everyone here is living the good life. Blah. 
June 14th, 2018, 04:30 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390 
A far more interesting question is what is the maximum area of a triangle given some properties that limit it's area. For example what is the maximum area of a triangle with perimeter P. Or what is the maximum area given a triangle with two sides of length p and q. 
July 2nd, 2018, 07:05 PM  #5 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 661 Thanks: 87 
Using the semiperimeter formula for the area of a triangle given its sides that I learned here, I think the greatest possible angle with a given perimeter is an equilateral triangle. I don't know if it's possible to prove this by setting a derivative equal to 0 similar to the proof that the given a fixed perimeter the rectangle with the largest area is a square. If the greatest possible area is an equilateral triangle, the maximum area with a perimeter of 3 is all three sides p/3, and the maximum area with sides p and q is p = q = the other side.


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