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

ANSWER

The area of the shaded portion is 104 in²

EXPLANATION

The total area of the metal is:

$A_{metal}=18\times6=108in$

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:

$A_{\text{triangle}}=\frac{b\times h}{2}=\frac{b\times b}{2}=\frac{b^2}{2}$

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

$\begin{gathered} 2^2=b^2+h^2 \\ 2^2=b^2+b^2^{} \\ 4=2b^2 \\ b^2=\frac{4}{2} \\ b^2=2 \end{gathered}$

Replacing into the expression for the area of one