Question

a rectangular metal has a length of 13 inches and a width of 6 inches. A 45-45-90 triangle is cute from each corner. The hypotenuse of each triangle is 2 inches. What expression can solve for the area of the shaded portion in square in.

Answer 1

The area of the shaded portion is 104 in²

Step-by-step explanation

The total area of the metal is:

`A_(metal)=18*6=108in`

To find the area of the shaded portion we have to subtract the areas of the triangles.

Each triangle is a right isosceles triangle - because there are two angles with the same measure, 45º - so we can say that the base and the height of the triangles are the same:

`b=h`

The area of one triangle is:

`A_{\text{triangle}}=(b* h)/(2)=(b* b)/(2)=(b^2)/(2)`

By the pythagorean theorem, the base of each triangle is:

`\begin{gathered} 2^2=b^2+h^2 \\ 2^2=b^2+b^2^{} \\ 4=2b^2 \\ b^2=(4)/(2) \\ b^2=2 \end{gathered}`

Replacing into the expression for the area of one