Expert: Ahmed Salami - 8/6/2009

Question
Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3cm. & 4cm. if two sides of the rectangle lie along the legs.

Answer
Hi Brian,
It makes it easier if we use a coordinate system involving the x and y axes for this problem. Consider two sides of the rectangle to lie on the x and y axes. We now have vertices on these axes and one at the origin. The last vertex would lie on the right triangle. We need to find the relationship between x and y at this point.
Now, consider the hypotenuse to be a line on this coordinate system, the equation of the line is gotten from the formula
x/a + y/b = 1
where a and b are the x and y intercepts respectively
If we take the shorter side to lie on the x axis then a = 3 and b = 4. The equation of the line is then
x/3 + y/4 = 1
y/4 = -x/3 + 1
y = -4x/3 + 4
The area of the rectangle is
A = xy
 = x(-4x/3 + 4)
 = -4x²/3 + 4x
We now need to find the value of x that gives the maximum area, to do this we find dA/dx and equate to zero.
dA/dx = -8x/3 + 4
equating to zero,
-8x/3 + 4 = 0
4 = 8x/3
x = 3/2
y = (-4/3)(3/2) + 4
 = -2 + 4 = 2
The largest area is then
A = (3/2).2
 = 3cm²