The problem states that in the diagram, $EF = 8$ cm, $FG = x$ cm, and $GH = (x+2)$ cm. Also, $\angle EFC = 90^{\circ}$. The area of the shaded portion (triangle $EGH$) is 40 $cm^2$. We need to find the area of triangle $EFG$.

1. Problem Description

The problem states that in the diagram, $EF = 8$ cm, $FG = x$ cm, and $GH = (x+2)$ cm. Also, $\angle EFC = 90^{\circ}$. The area of the shaded portion (triangle $EGH$) is 40 $cm^2$. We need to find the area of triangle $EFG$.

2. Solution Steps

First, we need to find the area of triangle $EGH$ in terms of $x$. The base of the triangle is $GH = x+2$, and the height of the triangle is $EF = 8$ cm. Since the area of the shaded portion (triangle $EGH$) is 40 $cm^2$, we can write:

$\frac{1}{2} \cdot (x+2) \cdot 8 = 40$

Simplify the equation:

$4(x+2) = 40$

$x+2 = 10$

$x = 8$

Now that we have the value of $x$, which represents the length of $FG$, we can calculate the area of triangle $EFG$. The base of triangle $EFG$ is $FG = x = 8$ cm, and the height is $EF = 8$ cm. The area of a triangle is given by:

$Area = \frac{1}{2} \cdot base \cdot height$

Area of triangle $EFG$ = $\frac{1}{2} \cdot FG \cdot EF = \frac{1}{2} \cdot 8 \cdot 8 = \frac{1}{2} \cdot 64 = 32$

3. Final Answer

The area of triangle $EFG$ is 32 $cm^2$.