



What is
the largest rectangle that can be drawn inside a right triangle



with
sides of 9 and 12 units and hypotenuse of 15 units? 





Before
you do anything else, draw a picture of the problem. 








OK, the
way to solve a problem like this 


is to
place it on an X, Y graph. 


Put the
right triangle at the origin (0,0) point. 





The trick
to solving this one is to use the equation of the line 


that is
the hypotenuse of the triangle. 


We know
two points on that line: (0,9) and (12,0). 


With
these two points, we can find the equation of the line. 


First we
need to find the slope of the line. 


The line
goes down from left to right 


so the
slope will be a negative number. 








The other
thing we need to know 


is the
point where the line crosses the Y axis. 


That is,
the point where X = 0. 


We don't
need to look too hard for that point, It's at (0,9)! 





The slope
intercept form of the equation of a line looks like this: 





Y
= mX + b






And for
our line, b = 9 and m = ^{3}/4. 


That
means the equation of the line is: 











This is
the equation of the hypotenuse line. 


Every
point on the hypotenuse has an X, Y value 


that
makes the equation true. 


THAT
INCLUDES THE POINT THAT IS THE CORNER OF THE RECTANGLE!!! 





We can
call that point, whatever it is, (X, Y) 


But we
also know that the point is on the line: 











So
instead of using the name Y, and calling the point (X, Y), 


We can
call it: 








We can
also name the points on the other three corners of the rectangle: 








Now the
last trick. 


From left
to right, 


the
rectangle goes from an X value of 0 to a value of X. 


So the
width of the rectangle is X. 


From
bottom to top, 


the
rectangle goes from a value of Y = 0 to Y = ^{3}/4X + 9. 


So the
height of the rectangle is ^{3}/4X + 9 





THAT WAS
THE BIG TRICK OF THIS PAGE. 


DID YOU
CATCH IT? 


WE WERE
ABLE TO WRITE Y IN TERMS OF X 








So the
area equation is: 








Well what
do you know? An upside down "U"! 








So the
maximum area happens when X = 6, What is Y? 








So: 








copyright 2005 Bruce Kirkpatrick 
