Calculate Area of House Shape Using Expressions 12x+9 and x+2y

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:

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

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