Calculate Rectangle Perimeter: Area Ratio 1/3 Between Triangle and Rectangle

Let's first look at triangle BCE and calculate side EC using the Pythagorean theorem:

$BC^2+EC^2=BE^2$

Let's substitute the known values:

$6^2+EC^2=10^2$

$36+EC^2=100$

$EC^2=100-36$

$EC^2=64$

Let's find the square root:

$EC=8$

Let's calculate the area of triangle BCE:

$S=\frac{BC\times EC}{2}$

Let's substitute the known values:

$S=\frac{6\times8}{2}=\frac{48}{2}=24$

According to the given data, the area of triangle BCE is one-third of rectangle ABCD's area, therefore:

$24=\frac{1}{3}$

Let's multiply by 3:

$S=3\times24=72$

The area of the rectangle equals 72

Now let's find side CD

We know that the area of a rectangle equals length times width, meaning:

$S=BC\times DC$

Let's substitute the known values in the formula:

$72=6\times CD$

Let's divide both sides by 6:

$CD=12$

Since in a rectangle opposite sides are equal, AB also equals 12

Now we can calculate the perimeter of rectangle ABCD:

$12+6+12+6=24+12=36$