# Find the area of the shaded region $DFEB$

$\Delta ABE$ and $\Delta DBC$ overlapped as shown below. Given that $\angle ABC=90 ^{\circ}$, $AD=2cm$ , $DB=6cm$, $BE=8cm$ and $EC=4cm$.

Finding the area of each triangle was easy, I got $\Delta ABE=32cm^2$ and $\Delta DBC=36cm^2$. From the figure we can see
-$\angle ADF$ and $\angle BDF$ are supplementary angles (1) so are
*$\angle AFD$ and $\angle EFD$ (2)
*$\angle EFC$ and $\angle EFD$ (3)
*$\angle BEF$ and $\angle CEF$ (4).
-$\angle AFD$ and $\angle CFE$ are opposite angles

Also I tried to do some takeaways of the triangles but it does not seem to work.

How should I approach this type of geometric problems?

• the problem is quite easy – Marvel Maharrnab Jul 8 '17 at 15:37
• should i give hints or solve it – Marvel Maharrnab Jul 8 '17 at 15:38
• Hints following elementary math would be great! – User 2524 Jul 8 '17 at 15:41

Hint:

use a coordinate system centered at $B$ such that: $$A=(0,8)\quad D=(0,6) \quad E=(8,0) \quad C=(10,0)$$ and $F$ is the intersection point of the two lines: $$\frac{x}{8}+\frac{y}{8}=1 \qquad\frac{x}{10}+\frac{y}{6}=1$$

The coordinates of $F$ are the heights of the two triangles $ADF$ and $ECF$.

Let F = (h, k).

$[red] = \dfrac {(8)(k)}{2}= 4k$. Other areas are similarly calculated.

Note that $h + 3h + 4k = [brown] + [green] + [red] = [\triangle ABE] = \dfrac {(8)(8)}{2}$.

That is, $h + k = 8$ …. (1)

Similarly, we have, $h + 2k = 12$ …. (2)

Result follows by solving (1) and (2).

Drop a perpendicular line from $F$ on $BC$ that meets it at $G$. Since $AB=BE$, so $\angle FEG=45^\circ$ and therefore $\angle GFE=45^\circ$. Thus $FG=GE=x$ say. Then in the triangles $FGC$ and $DBC$, $$\frac{x}{x+4}=\frac{6}{12}\Rightarrow x=4cm.$$ Now the required area is the sum of the area of trapezoid $DFGB$ and triangle $FGE$ which are readily calculated.

Area of yellow+ orange region=A{use your brains}

Area of yellow+ orange +green region=B

Area of orange region={Area of yellow+ orange +green region}-{Area of yellow+ orange region}=A-B

Area of blue+ orange region=C{use your brains}

Area of blue+ orange +green region=D

Area of green region={Area of blue+ orange +green region}-{Area of blue+ orange region}=C-D

Area of green+orange region=(A-B)+(C-D)