Find the area of the shaded part?

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3 Answers
Sep 26, 2016

#32 "cm"^2#

Explanation:

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#.

Sep 27, 2016

See below.

Explanation:

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.

Oct 3, 2016

#color(red)("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:

#1/2xx" base" xx "height"#

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

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

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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")#