Maximum area of triangle

[Videoorchard]

Hi,

The lenghths of the two shorter legs of right triangle add up to 40 units. what is the maximum possible area of triangle?

For the above problem the author states that: all right triangles with a given perimeter, the isosceles right triangle has the greatest area..My question is where is the perimeter(sum of all the 3 sides) declared? We only know sum of 2 sides. i.e 40. Should i consider the perimeter i.e 80 formed by the rectangle? whats happening?


Question:
Geometry guide
Chapter 4, problem set
Question 18.

[n00bpron00bpron00b]

Given -

a) We've been told the triangle is an right triangle

b) Sum of two short legs of the right triangle = 40 (i.e not the hypotenuse but the other two legs)

c) Find the max. area of triangle

My question is where is the perimeter(sum of all the 3 sides) declared? We only know sum of 2 sides.

-> We don't need to know the length of the third side (i.e the hypotenuse)

-> We don't need to know the total perimeter of the right triangle

Find the max. area of triangle

Area = 1/2 * base * height

The two short legs of the right triangle will form the base and height (any side could be the base and any side could be the height ; does not matter)

Okay, what if we fail to recognize that the area of an isosceles right triangle is maximum

If sum of lengths of the two short sides = 40
Let's say, one side is 20 and other side is 20 (total = 40)
assign one side as base and other side as height
Area = 1/2 * 20 * 20 = 400/2 = 200

Now check for one more condition
if length of one side is 21 and the length of other side is 19 (total =40)
Area = 1/2 * b * h = 1/2 * 21 *19 = 199.5

We can generalize, for any length values besides (20,20) ; you will notice that one length becomes less than 20 and the other length is greater than 20 (for total length to become 40) => and for such values the area will always be less than 200

200 is the max area

[Videoorchard]

Hi,

Thank you for an clear explanation noob. Helped me!

For people reading this post, Few properties which may help you regarding maximum question in geometry

#1. Area of an isosceles right triangle is maximum. (Example above)

#2 : For a given length, the largest area is enclosed by a circle.

#3. If you are given 2 sides of a triangle or a parrallelogram, you can maximize the area by placing the 2 sides perpendicular to each other.

#4. A regular polygon with all sides equal has:
1. Max area for a given perimeter
2. Minimum perimeter for a given area

P.S
The #4 mainly applies to square and and equilateral triangle..

[tommywallach]

Great rules video. Thanks for posting! A couple quick things:

#1. Area of an isosceles right triangle is maximum. --> Area of a right triangle with a given limited perimeter is maximized as an isosceles right triangle.

#2 : For a given length, the largest area is enclosed by a circle. --> For a given perimeter that can enclose any shape, the enclosed area will be larger the more sides you have, and the largest area will be enclosed by a circle (i.e. infinite sides).

#4. A regular polygon with all sides equal has Max area for a given perimeter --> A polygon with a defined number of sides and a defined perimeter will have the max area if all sides are equal (i.e. it is regular).

Thanks!

-t