My Math Forum Max. area of an inscribed rectangle

 Algebra Pre-Algebra and Basic Algebra Math Forum

 June 20th, 2012, 02:21 AM #1 Member   Joined: Jan 2012 Posts: 52 Thanks: 0 Max. area of an inscribed rectangle Hello, How do I find the maximum area (the biggest value of the area) of a rectangle in the following case: Be a, b, c the sides of a right triangle such the $a^2= b^2 + c^2$ If x and y are the sides of a rectangle such as x belongs to the side b and y to the side c of triangle and one vertex of the rectangle touches the side a, find the maximum value for x.y in terms b and c. Thanks!
 June 20th, 2012, 04:49 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond Re: Max. area of an inscribed rectangle Under such a constraint, the largest rectangular shape, in terms of area, is a square, so x = y. Using the formula for the area, A, of a triangle we have A = bc/2. Sketching a diagram we observe that the area of the triangle can be written as $\frac{(b\,-\,x)x}{2}\,+\,\frac{(c\,-\,x)x}{2}\,+\,x^2\,=\,\frac{bc}{2} \\ \Rightarrow\,x\,=\,\frac{bc}{b\,+\,c}$ So the greatest area is $$$\frac{bc}{b\,+\,c}$$^2$.
June 20th, 2012, 05:04 AM   #3
Member

Joined: Jun 2012
From: UK

Posts: 39
Thanks: 0

Re: Max. area of an inscribed rectangle

Quote:
 Originally Posted by greg1313 Under such a constraint, the largest rectangular shape, in terms of area, is a square.
I beg to differ.

Largest area, bc/4, occurs when x=b/2 and y=c/2.

 June 20th, 2012, 05:56 AM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond Re: Max. area of an inscribed rectangle You're right - again. I had bc/4 as a solution, but discarded it after an erroneous calculation.

 Tags area, inscribed, max, rectangle

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post guru123 Algebra 2 May 19th, 2012 10:36 PM jaredbeach Calculus 4 January 10th, 2012 06:47 PM simranjit Algebra 5 November 28th, 2010 10:12 PM magician Algebra 1 October 12th, 2009 03:33 AM djodyssey1127 Algebra 3 January 22nd, 2008 10:18 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top