Which side of the triangle gives it maximu area?

In summary, the conversation discussed how to find the maximum area of an isosceles triangle with two given side lengths. The participants considered using squares and equilateral triangles, but ultimately determined that a right triangle would have the greatest area. They also discussed using trigonometric ratios and calculus to find the height of the triangle in terms of the base length. Finally, they concluded that the best way to find the maximum area is by flipping the triangle and pivoting the given side to get the most height possible.
  • #1
MathLete
For my homework (actually its a bonus question) I was given an isocelese triangle with 2 sides lengths defined. Let's call it triangle ABC.

Code:
    A
    /\
   /  \
  /    \
 /      \
/--------\
B         C
side AB is given 5 and side AC is also 5. I have to give BC a length for the triangle to have the maximum area.

First I thought of squares. A square has more area then a rectangle with the same parameter. 5x5 is greater than 1x20 thus proves my point of squares. I thought that If the triangle would be equalateral it would have the greatest area. But I was wrong after expirementing with numbers.

Then I thought since a square has the greatest area then half a square would be a triangle and therefore the right angle triangle should have the greatest area. Witch would have sqrt(50) as BC and an area of 12.5 according to my calculations (may be wrong).

Any thoughts on what kind of triangle will have the greatest area?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Can you write a formula for the area of the triangle in terms of the length of the base? That might suggest a way to find the maximum area...
 
  • #3
I don't know what you mean by that. the area of a triangle is base times hight divided by 2(bxh/2). You can use trig rations and sine/cos laws to find the hight of the triangle.
 
  • #4
Well do that then; find the height of the triangle in terms of the length of the base.
 
  • #5
If you're lazy, you can use some trigonomery.
S=(1/2)bcSinA
Which angle has the biggest sine? pi/2 (+k2pi).
 
  • #6
Just use some simple calculus (or cheat and just graph it on your calculator).

Look at this picture here
http://myfiles.dyndns.org/pictures/triangle1.jpg

Now based on that diagram I can make 2 formulas:

[tex]A = xy[/tex] since x is only half the base here

[tex]y = (5^2 - x^2)^\frac{1}{2}[/tex]

Combine them to get this

[tex]A = x(5^2 - x^2)^\frac{1}{2}[/tex]

Now differentiate it

[tex]\frac{dA}{dy} = (5^2 - x^2)^\frac{1}{2} - \frac{y^2}{(5^2 - x^2)^\frac{1}{2}}[/tex]

Make that formula equal to 0 and you get x.
Since my diagram had x as only HALF of the base length, double what you get for x.

I get the base as being 7.071 long. That gives the triangle and area of 12.5


Now about what kishtik said
If you're lazy, you can use some trigonomery.
S=(1/2)bcSinA
Which angle has the biggest sine? pi/2 (+k2pi).

That gives the exact same area. (1/2)(5)(5) = 12.5

Give the teacher both answer and you might get extra bonus marks
 
Last edited by a moderator:
  • #7
This isn't that hard. Flip the triangle so that a given side is on the bottom. (Base) You know that the formula for area is base x height x 1/2.

Now pivot the other given side until you get the most height out of it. The base is given, you have the most height you can possibly get in ONE position only. Is this sinking in?
 

1. What is the maximum possible area of a triangle?

The maximum possible area of a triangle depends on the length of its sides. The larger the length of the sides, the larger the area will be.

2. How do you find the area of a triangle with given side lengths?

To find the area of a triangle with given side lengths, you can use Heron's formula: A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter of the triangle and a, b, and c are the lengths of the sides.

3. How do you determine which side of a triangle gives it maximum area?

The side length that gives a triangle maximum area is the longest side. This is because the area of a triangle is maximized when the triangle is equilateral, and the longest side is also the side with the largest angle, making the triangle more equilateral.

4. Can a triangle have infinite area?

No, a triangle cannot have infinite area. The area of a triangle is always finite and is determined by the length of its sides.

5. How can I find the maximum area of a triangle with only one side given?

If you only have one side of a triangle given, you cannot determine the maximum area. You need at least two side lengths to calculate the area of a triangle using Heron's formula or other methods.

Similar threads

Replies
16
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
870
  • Introductory Physics Homework Help
Replies
9
Views
874
  • Precalculus Mathematics Homework Help
Replies
5
Views
1K
Replies
1
Views
732
  • Introductory Physics Homework Help
Replies
3
Views
114
  • Precalculus Mathematics Homework Help
Replies
9
Views
1K
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
803
Replies
12
Views
2K
Back
Top