Area of a Rectangle: Extended distributive law

Let's begin by calculating the area of the rectangle using the given data:

$64=4x\times x$

$64=4x^2$

Next we will divide both sides by 4:

$16=x^2$

And we will remove the square root:

$4=x$

Therefore, BC equals 4.

Since in a rectangle each pair of opposite sides are equal to each other, let's call each pair of sides X and Y

Now let's set up a formula to calculate the perimeter of the rectangle:

$2x+2y=14$

Let's divide both sides by 2:

$x+y=7$

From this formula, we'll calculate X:

$x=7-y$

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

$x\times y=12$

We know that X equals 7 minus Y, let's substitute this in the formula:

$(7-y)\times y=12$

$7y-y^2=12$

$y^2-7y+12=0$

$(y-3)\times(y-4)=0$

From this we can claim that:

$y=3,y=4$

Let's go back to the formula we found earlier:

$x=7-y$

Let's substitute y equals 3 and we get:

$x=7-3=4$

Now let's substitute y equals 4 and we get:

$x=7-4=3$

Therefore, the lengths of the rectangle's sides are 4 and 3

Let's draw a line in the middle of the drawing that divides the house into 2

Meaning it divides the triangle and the rectangular part.

The 2 lines represent the heights in both shapes.

If we connect the height of the roof with the height of the rectangular part, we get the total height

Let's put the known data in the formula:

$\frac{1}{2}h_{\text{square}}+h_{square}=12x+9$

$\frac{3}{2}h_{\text{square}}=12x+9$

We'll multiply by two thirds and get:

$h_{\text{square}}=\frac{2(12x+9)}{3}=2(4x+3)$

$h_{\text{square}}=8x+6$

If the height of the triangle equals half the height of the rectangular part, we can calculate it using the following formula:

$h_{\text{triangle}}=\frac{1}{2}(8x+6)=4x+3$

Now we can calculate the area of the triangular part:

$\frac{(x+2y)\times(4x+3)}{2}=\frac{4x^2+3x+8xy+6y}{2}=2x^2+1.5x+4xy+3y$

Now we can calculate the rectangular part:

$(x+2y)\times(8x+6)=8x^2+6x+16xy+12y$

Now let's combine the triangular area with the rectangular area to express the total area of the shape:

$S=2x^2+1.5x+4xy+3y+8x^2+6x+16xy+12y$

$S=10x^2+20xy+7.5x+15y$