Triangle question

[Lisa1]

Hi I am going to follow the above methods after work today ,ill post my solutions on here so you can see it and correct me if im wrong anywhere...eager to get to result. Thank you

 

[Dr.Peterson]

Using the Doctor’s approach

“ make two sides 2 and 3 and allow the third side to vary in length “

you could let third side = x.

Do you think you could use Heron’s formula to first write semiperimeter(s) in terms of x and then following substitution express A² in terms of x?

After simplifying you should end up with

A² = B( )( )where B represents a fraction and each set of brackets holds a quadratic expression which is a difference of two squares.

The next part you have to think about carefully. If one set of brackets represents a length and the second set of brackets represents a width what shape would give the maximum value for A² and therefore maximum area?

How do the two quadratic expressions relate to each other?

Once you know the answer to this you’re almost there. See if you can finish off.

There's a much simpler way to approach it; you can start by just thinking visually about, say, making 3 the base and considering which position for the 2 will yield the greatest area using the formula A = bh/2. Or the same could be done algebraically, ignoring the third side for the moment and instead varying something else, like the angle.

 

[Lisa1]

Hi
I am struggling with this , is there any chance you can post apossible solution for this please ? Will be helpful

 

[hoosie]

Some extra help for you Lisa:

Heron’s Formula:



where s = half triangle’s perimeter

s = (a+b+c)/2 where a = 3, b = 2, c = x
s = (x + 5)/2

A² = s(s - 3)(s - 2)(s - x)
= (s² - 3s)(s² - xs - 2s + 2x)
= [(1/4)(x + 5)² - 3(x + 5)/2][(1/4)(x + 5)² - x(x + 5)/2 - 2(x + 5)/2 + 2x]

Can you continue?

 

[Dr.Peterson]

Hi
I am struggling with this , is there any chance you can post apossible solution for this please ? Will be helpful

Here's my approach: given two sides of a triangle, the area will be maximized when they are perpendicular, since for a fixed base you want the maximum height. Then you just have to check that the third side, the hypotenuse, is not longer than it is allowed to be. You can repeat this for each pair of sides, then ponder.