Maximum area of a triangle inscribed in another triangle?

In summary, to maximize the area of BDE, the length of DE should be (a/2). To maximize the perimeter of BDE, the length of DE should be (a/2) + (a-a/x).
  • #1
Daveigh
5
0

Homework Statement


[/B]
Hello!

I have this question which I don't quite know how to solve...

ABC is an equilateral triangle - the length of its sides equal to (a).
DE is parallel to BC

1. What length should DE be to achieve the largest possible area of triangle BDE?
2. What length should DE be to achieve the smallest possible perimeter of triangle BDE?


How should this be done (step-by-step, my knowledge of math is quite basic)?

RrU2DrM.jpg


The attempt at a solution

I tried to solve the first question and got that DE should be (a/2) to make the maximum area of BDE. (I let DE = (a-a/x) and BD = (a/x), I used a formula to calculate area of a triangle using 2 sides and the angle between (120°) and then I derived the area expression...
But I think that what I was attempting is very messy, and I am unsure if the way I did it is correct.

Also, I didn't manage to solve the second question about the perimeter.
If anyone could explain and show the correct answer for comparison, I would be very grateful.
 
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  • #2
Daveigh said:
I tried to solve the first question and got that DE should be (a/2) to make the maximum area of BDE. (I let DE = (a-a/x) and BD = (a/x), I used a formula to calculate area of a triangle using 2 sides and the angle between (120°) and then I derived the area expression...
But I think that what I was attempting is very messy, and I am unsure if the way I did it is correct.
The method looks fine and the answer is correct, that looks good.
It is easier if you use the formula A=1/2 (DE)*(height) because then you avoid dealing with odd angles.

Daveigh said:
Also, I didn't manage to solve the second question about the perimeter.
What did you get as perimeter?
 
  • #3
I tried making an expression for the perimeter but when I differentiated it it was so large and I didn't know what to do next.
(a/x) + (a-a/x) + sqrt((a/x)^2+(a-a/x)^2-2*(a/x)*(a-a/x)*cos120°)
I couldn't get rid of 'a' to find 'x'...
 
  • #4
Daveigh said:
I tried making an expression for the perimeter but when I differentiated it it was so large and I didn't know what to do next.
(a/x) + (a-a/x) + sqrt((a/x)^2+(a-a/x)^2-2*(a/x)*(a-a/x)*cos120°)
I couldn't get rid of 'a' to find 'x'...

You can factor out a from the expression of the perimeter. Note that a/x + (a-a/x) = a.
Expand the parentheses under the square root and simplify before taking the derivative.
 
Last edited:
  • #5
There are two other tricks that can help:
The minimum of f(x) is also a minimum of f(x)+b with some constant b.
The minimum of f(x) is also a minimum of f2(x). The minimum of f2(x), if not zero, is a minimum OR maximum of f(x) depending on the sign of f(x).

But there is an even better approach: ehild's hint gives you the sum of two sides of the triangle, and it is constant. You just have to minimize the length of a specific side (which one?), and this can be done without calculations.
 

Related to Maximum area of a triangle inscribed in another triangle?

1. What is the maximum area of a triangle inscribed in another triangle?

The maximum area of a triangle inscribed in another triangle is half of the area of the larger triangle. This is known as the "half base times height" formula.

2. How do you find the maximum area of a triangle inscribed in another triangle?

To find the maximum area, you need to first find the length of the base and the height of the inscribed triangle. Then, you can use the formula "1/2 * base * height" to calculate the maximum area.

3. Can the maximum area of a triangle inscribed in another triangle be greater than half of the area of the larger triangle?

No, the maximum area of a triangle inscribed in another triangle can never be greater than half of the area of the larger triangle. This is because the inscribed triangle is always a fraction of the larger triangle.

4. What is the significance of finding the maximum area of a triangle inscribed in another triangle?

Finding the maximum area of a triangle inscribed in another triangle is useful in various applications, such as optimizing the use of space in a given area. It can also help in solving geometric problems and proving certain theorems.

5. Are there any other ways to find the maximum area of a triangle inscribed in another triangle?

Yes, there are other methods such as using the Pythagorean theorem or trigonometric ratios to find the length of the base and height of the inscribed triangle. However, the "half base times height" formula is the most commonly used method.

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