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

Observe the following rectangle:

The the area of the triangle ΔBCE is$\frac{1}{3}$ the area of the rectangle ABCD.

Calculate the perimeter of the rectangle ABCD.

Observe triangle BCE and proceed to calculate side EC using the Pythagorean theorem:

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

Insert the known values into the theorem:

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

$36+EC^2=100$

$EC^2=100-36$

$EC^2=64$

Determine the square root:

$EC=8$

Calculate the area of triangle BCE:

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

Insert the known values once again:

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

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

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

Multiply by 3:

$S=3\times24=72$

The area of the rectangle equals 72

Now let's determine side CD

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

$S=BC\times DC$

Insert the known values in the formula:

$72=6\times CD$

Divide both sides by 6:

$CD=12$

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

Proceed to calculate the perimeter of the rectangle ABCD:

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