MATHCOUNTS Mini #45 -- Maximum Area of Inscribed Rectangles & Triangles

This video focuses on the maximum area of inscribed rectangles & triangles.

Video by Art of Problem Solving's Richard Rusczyk, a MATHCOUNTS alum. Visit Art of Problem Solving for many more educational resources.

[MrKrabs-xftr]

A good formula to always know is that the area of the largest rectangle able to be inscribed within a triangle is always half the area of the triangle.

[Philipel]

That was a really clever way to show that the rectangle was the largest possible

[michaelempeigne]

Area of triangle using heron's formula = sqrt [ s ( s - a ) ( s - b ) ( s - c ) ], where s = ( a + b + c ) / 2

so s = 24 and Area = sqrt [ 24*14*7*3 ] = 84

84 = ( 1 / 2 ) * 21 * h
(84 / 21) * 2 = h
h = 8

[mateiacd]

Excelent idea, thank you ! I think it would have helped if you had assigned letters A, B, C to the vertices of the triangle; D, E, F, G to the vertices of the inscribed rectangle; I, J to the other two points on the base resulted from flipping at 08:40; H to the intersection of those flipped sides

[richardxu]

By algebra we can get the same answer. Say the triangle has length a, b and c with one side of the rectangle with length x on c, and the other side of the rectangle equals d, the height of the triangle on side c equals h, x=c(h-d)/h, S=c(h-d)/h*d/2 =c/h((h-d)h/2). c/h is constant, so S is the biggest when (h-d)h is the biggest.
(h-d)h=-(d-h/2)^2+1/4*h^2, so the rectangle is the biggest when d=h/2

[MKWKezer]

Really elegant solution, I love it. :)

[jwm]

...A splendid demo of a concept of calculus...the thinner and thiner rectanlges, then adjusting their sizes to show why a particular shape can have maximum area.  Are MathCounts {tm} students axquainted with Heron's Formula?

[keshavb]

I found the other guy finding the area of a rectangle with sides 6.9 and 10 with calculus.
He used coordinate geometry and got a limit as n→infinity of something and got 69 the long way.

[howardtang]

How do you know that if the side is too short that the top piece will go outside of the triangle?

[aashimasinghsisodia]

How can we use Heron’s formula in this? I see a lot of comments about

[howardtang]

How do you know if one side is smaller that the area will be smaller? Couldn't the other side affect the area as well?

[vnsomeshwar]

I used Heron's formula but I accidentally used the full perimeter instead of the semiperimeter in the problem. Whoops! :)

[abdixsimplix]

I used an alterative method consisting of a derivative to find maximum.

[richardxu]

If the triangle is not obtuse, the three cases will result in the same answer, area of the triangle/2.

[keshavb]

there's a much faster method to find area of triangle, a rectangle minus the 6-8-10 and 5-8-17 triangles.

[richardxu]

What happened in the end? The last sentence after the solution.

[dennisreed]

How would you do this all(go through the though process) in the sprint time period?

[aakashSky-]

can some plz tell me how to derive the formula for Counting Rectangles in an n X m figure.

[Deathranger]

Or you could've just used Heron's Formula. :P A whole lot easier IMO.