Area of a Triangle: Extended distributive law

We use the formula to calculate the area of the right triangle:

$\frac{AC\cdot BC}{2}=\frac{cateto\times cateto}{2}$

And compare the expression with the area of the triangle $6$

$\frac{4\cdot(X-1)}{2}=6$

Multiplying the equation by the common denominator means that we multiply by $2$

$4(X-1)=12$

We distribute the parentheses before the distributive property

$4X-4=12$ / $+4$

$4X=16$ / $:4$

$X=4$

We replace $X=4$ in the expression $BC$ and

find:

$BC=X-1=4-1=3$

To solve this problem, we will calculate the length of side $FE$ using the area formula for a triangle:

• Step 1: Use the formula for the area of a triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$.

• Step 2: Substitute the given values into the formula.
  We know that $A = 24 \, \text{cm}^2$ and $\text{height} = 8 \, \text{cm}$.

• Step 3: Set up the equation: $24 = \frac{1}{2} \times \text{base} \times 8$.

• Step 4: Simplify and solve for the base:$24 = \frac{8}{2} \times \text{base} \rightarrow 24 = 4 \times \text{base}$.

• Step 5: Solve for $\text{base}$: $\text{base} = \frac{24}{4} = 6 \, \text{cm}$.

Therefore, the side $FE$ of the triangle is 6 cm.

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 obtain the total height.

Let's insert 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 as follows:

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

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

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:

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

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