Find the curve of minimum length which divides the equilateral triangle into two pieces of equal area (median curve)

At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get our expert's

answer on brainly

SEE EXPERT ANSWER

Get your free account and access expert answers to this and thousands of other questions.

A community for students.

Find the curve of minimum length which divides the equilateral triangle into two pieces of equal area (median curve)

Mathematics
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your free account and access expert answers to this and thousands of other questions

|dw:1435819109185:dw|
This must be fairly easy with variational methods but I'm wondering if there's a more elementary method that only uses calculus. I think I can sort of see that at every point the curve must be perpendicular to the line segment connecting it to the vertex, but I wouldn't know how to prove it.|dw:1435826489623:dw| The image sort of describes my intuition. Since over the small distance r doesn't change, the biggest area we can get is by going perpendicular to r. I guess this would imply that we need to go along a circle centered on the facing vertex.
"the biggest area we can get is by going perpendicular to r." I meant : the largest amount of area we can get for a given additional length ds.

Not the answer you are looking for?

Search for more explanations.

Ask your own question

Other answers:

so i cant make only lne ?
You have a very good imagination @beginnersmind ! For simplicity, I guess we may use the fact that of all the closed curves of given area, a circle has minimum perimeter.
what type of circle ?
@ikram002p median of a triangle does divide the triangle into two parts of equal area, but it need not be of minimum length right
|dw:1435827736653:dw|
or |dw:1435827776826:dw|
yes i think beginnersmind is referring to the second one
well we always can construct a short cut for curves |dw:1435827828432:dw|
Yeah, that one. I get 0.6734 for length, while using a line takes 0.707 (1 over squareroot 2)
"You have a very good imagination @beginnersmind ! For simplicity, I guess we may use the fact that of all the closed curves of given area, a circle has minimum perimeter." Thanks. I think I got it now. Let's assume there's another curve with a shorter length that encloses the same amount of area. Then we could copy it 5 more times to create a closed curve which has the same area as a circle but smaller perimeter, which would be a contradiction.|dw:1435828385612:dw|
If u wanna the smallest closed curve inside a triangle that touches its edges that would be a circle.
Excellent! that gives us a closed curve inside a hexagon. The closed curve divides the hexagon into two parts of equal area
Lovely <3
Really cool puzzle. Seems almost impossible at first but actually all you need is a good idea and the fact about the circle having the smallest perimeter for a given enclosed area.
|dw:1435915373490:dw|

Not the answer you are looking for?

Search for more explanations.

Ask your own question