# Find the area of the shaded part?

Sep 26, 2016

$32 {\text{cm}}^{2}$

#### Explanation:

The area $A$ of a triangle with base $b$ and height $h$ is given by $A = \frac{1}{2} b h$.

If we treat the side of length $8 + 4 = 12$ as a base of the large triangle, then as the line with length $6$ forms a right angle with that side, the triangle has a height of $6$. Thus the area of the large triangle is $\frac{1}{2} \left(12\right) \left(6\right) = 36$.

Similarly, if we treat the length $4$ side of the white triangle as its base, then it has a height of $2$, meaning its area is $\frac{1}{2} \left(4\right) \left(2\right) = 4$.

As the area of the shaded section is the difference between the area of the large triangle and the area of the white triangle, we have our desired area as $36 - 4 = 32 {\text{cm}}^{2}$.

Sep 27, 2016

See below.

#### Explanation:

Supposing no tricks and using $A = \frac{b h}{2}$ we have

$A = {A}_{1} - {A}_{2} = \frac{\left(4 + 8\right) \times 6}{2} - \frac{4 \times 2}{2} = 36 - 4 = 32$

Now using Heron's formula with

$p = \frac{8 + 7 + 12}{2}$
${A}_{1} = \sqrt{p \left(p - 8\right) \left(p - 7\right) \left(p - 12\right)} \approx 26.91$

which is different from the former $36$. So the triangle's figure is a trick.

Oct 3, 2016

$\textcolor{red}{\text{Is this question correct in every detail?}}$

Area of the shaded portion ul("could be:")" " 32cm^2

#### Explanation:

Using the general principle that the area of a triangle is:

$\frac{1}{2} \times \text{ base" xx "height}$

The overall triangle area $\to \frac{1}{2} \times \left(4 + 8\right) \times 6 = 36 \textcolor{w h i t e}{.} c {m}^{2}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The smaller triangle area $\to \frac{1}{2} \times 4 \times 2 = 4 \textcolor{w h i t e}{.} c {m}^{2}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The area of the shaded portion -> (36-4) cm^2 = 32 cm^2" "????

$\textcolor{red}{\text{===================================}}$
$\textcolor{red}{\text{Checking a few things}}$

Using Pythagoras it should be the case that:

base of the whole $= 8 + 4 = 12 = \sqrt{{7}^{2} - {6}^{2}} + \sqrt{{8}^{2} - {6}^{2}}$

RHS $\to \sqrt{13} + \sqrt{28}$

$\sqrt{13} + \sqrt{28} \approx 8.9 \ne \text{ length of the base} = 12$

$\textcolor{red}{\text{Conclusion: There is contradicting information in the question}}$