Find the area of the shaded part?

The area `A` of a triangle with base `b` and height `h` is given by `A=1/2bh`.

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 `1/2(12)(6) = 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 `1/2(4)(2) = 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 "cm"^2`.

Supposing no tricks and using `A = (b h)/2` we have

`A = A_1 - A_2 = ((4+8)xx6)/2-(4 xx 2)/2 = 36-4=32`

Now using Heron's formula with

`p = (8+7+12)/2`
`A_1 = sqrt(p(p-8)(p-7)(p-12)) approx 26.91`

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

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

`1/2xx" base" xx "height"`

The overall triangle area `->1/2xx (4+8)xx6 = 36color(white)(.)cm^2`

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The smaller triangle area `-> 1/2 xx 4 xx 2 = 4color(white)(.)cm^2`

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

`color(red)("===================================")`
`color(red)("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 `-> sqrt(13)+sqrt(28)`

`sqrt(13)+sqrt(28) ~~8.9 !=" length of the base" = 12`

`color(red)("Conclusion: There is contradicting information in the question")`