Find the area of the shaded region

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In summary, the conversation revolves around finding the shaded region in a geometry problem. The participants discuss different methods and share their own attempts and solutions. The final solution involves using the concept of similar triangles and calculating the individual areas of the triangles to arrive at the total area of the shaded region. There is also a mention of using analytic geometry for the calculations.
  • #1
chwala
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Homework Statement
Consider the diagram below, find the area of the shaded region
Relevant Equations
similarity-congruency
1613261968943.png

this is another question that i saw on the internet...
 
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  • #2
1613262047990.png

This is my attempt, in the first attempt i tried to see whether i could be able to find the shaded region by way of calculations.
In the second attempt, i tried to use ratio,...
 
  • #3
i made a mistake in the area formula, the dimensions are not correct , let me re-do it...
 
  • #4
the dimensions ought to be ##3.72cm## and not ##1.5cm## that is the dimensions for each equilateral triangle...
 
  • #5
1613265416925.png


1613265458794.png


clearly this is not correct...i am missing something here...
 
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  • #6
AD should be 3.72 x 3.5.
 
  • #7
Lnewqban said:
AD should be 3.72 x 3.5.
Aaaaargh the last triangle...I will re do the working again...and at same time appreciate any alternative way of doing this.
 
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  • #8
chwala said:
Aaaaargh the last triangle...I will re do the working again...and at same time appreciate any alternative way of doing this.
I see 7 triangles, total area is 7 times 6 units of area: 42 units.
The diagonal line divides in half the shape formed by those 7 triangles: 21 units each big triangle.

Then, we remove the top triangle and investigate any symmetry among sides.
The top side measures 3 entire small-triangle sides.
In a symmetrical way, two of each truncated sides have the same dimension.

What could we do with that information?
Could we divide the area of the top big angle in two in order to obtain the value of the green area?
 
  • #9
Lnewqban said:
I see 7 triangles, total area is 7 times 6 units of area: 42 units.
The diagonal line divides in half the shape formed by those 7 triangles: 21 units each big triangle.

Then, we remove the top triangle and investigate any symmetry among sides.
The top side measures 3 entire small-triangle sides.
In a symmetrical way, two of each truncated sides have the same dimension.

What could we do with that information?
Could we divide the area of the top big angle in two in order to obtain the value of the green area?
Interesting...I am an analytical person, these geometry shapes and checking for congruence or similarity I don't like it :cool: :biggrin::biggrin:lol
 
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  • #10
chwala said:
Interesting...I am an analytical person, these geometry shapes and checking for congruence or similarity I don't like it :cool: :biggrin::biggrin:lol
Those are only two of the ways to reach the same conclusion. :smile:

If you also see 7 equilateral triangles, you would agree that the the diagonal line only devides the area of 6 of them.

Because in a symmetrical way, two of each truncated sides have the same dimension, we could imaginarily cut those shapes off from the big triangle formed by the 6 consecutive triangles and glue two matching sides together, obtaining three original triangles that way.

Therefore, the area of the green-colored parts is 6 units x 3 triangles = 18 units.
 
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  • #11
Lnewqban said:
Therefore, the area of the green-colored parts is 6 units x 3 triangles = 18 units.
Well, that's obviously wrong, or I read you wrong. Each of your two 'big triangles' is already 18 units. The green area encompasses a portion of that : looks like less than half.

(Not that I have anything positive to add).
 
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  • #12
chwala said:
Interesting...I am an analytical person, these geometry shapes and checking for congruence or similarity I don't like it :cool: :biggrin::biggrin:lol
I don't mean to judge. But surely you can take 5 min out of your day to understand the definition of similar triangles, there properties, and later similarity between different geometric objects? It will save you a lot of time this route, you learn more math, and spend less time using computations you don't need to do.
 
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  • #13
chwala said:
Interesting...I am an analytical person, these geometry shapes and checking for congruence or similarity I don't like it :cool: :biggrin::biggrin:lol

another member might have suggested this. but one could also set up a coordinate system and set up a handful of integrals.. not that anyone would WANT to do that lol
 
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  • #14
1613288526342.png

1613288561636.png

my other attempt...is the area of triangle ##AFT##=##4.396cm^2## or i am getting it wrong, are my angles correct?
 
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  • #15
MidgetDwarf said:
I don't mean to judge. But surely you can take 5 min out of your day to understand the definition of similar triangles, there properties, and later similarity between different geometric objects? It will save you a lot of time this route, you learn more math, and spend less time using computations you don't need to do.

noted, i will try boss...
 
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  • #16
hmmm27 said:
Well, that's obviously wrong, or I read you wrong. Each of your two 'big triangles' is already 18 units. The green area encompasses a portion of that : looks like less than half.

(Not that I have anything positive to add).
It was wrong, indeed.
Thank you.

After calculating the areas of different triangles, and using ratios among similar triangles, I have this new result:
The area of the green-colored parts is 4.5 + 2.0 + 0.5 = 7.0 units square.
 
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  • #17
Lnewqban said:
After calculating the areas of different triangles, and using ratios among similar triangles, I have this new result:
The area of the green-colored parts is 4.5 + 2.0 + 0.5 = 7.0 units square.
Would you provide some hints on what you did? I cannot get anywhere without resorting to analytic geometry.
 
  • #18
caz said:
Would you provide some hints on what you did? I cannot get anywhere without resorting to analytic geometry.
I used the similar triangles idea mentioned by @MidgetDwarf in post #12 above.

Please, see:
https://www.mathsisfun.com/geometry/triangles-similar.html

https://www.mathsisfun.com/algebra/trig-area-triangle-without-right-angle.html

ECB7628A-16C2-4B37-9A33-94EC25FD3C64.jpeg


Calculated the length of sides of any of the equilateral triangles.
##A=\sqrt{3}b^2/4=6##
##b=3.722~units##

Triangle AMN is similar to triangle ACD / Ratio = 4
Calculated length of CD
Calculated length of ##BC = b - CD##

Knowing that angle ABD is 60°, calculated area of triangle ABC, using
##A=(0.5)(AB)(BC)(sin~60)=4.5~units^2##

Triangles AGH and AKL are similar to triangle AMN.
Triangles EFG and IJK are similar to triangle ABC.

Repeated process for calculating areas of triangles EFG (##2.0~units^2##) and IJK (##0.5~units^2##).
Added individual areas for a total of ##7.0~units^2##.
 
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  • #19
Thanks guys...

i saw this answer below on the internet; I am trying to follow it...I thought i should share it here,
1613353840187.png


what i do not understand is "why is he squaring'
##\frac{A_1}{A_2}##=##\frac{0.75^2}{0.5^2}## is it really necessary to square?i do not think so,
MY understanding is that ##(ASF)=(LSF)^2##
therefore, ##LSF ## of ##\frac{A_1}{A_2}##=0.75## ×##2##=\frac{3}{2}##
from here it follows that ##(ASF)=\frac{9}{4}##
 
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  • #20
chwala said:
MY understanding is that
I don't know what you mean by ASF and LSF.
The area of similar triangles is proportional to the square of the length of corresponding sides.
 
  • #21
##(ASF)##= Area Scale Factor & ##LSF##= Linear Scale Factor...and of course,
##(VSF)##=Volume Scale Factor.
##(VSF)=(LSF)^3##
 
  • #22
i was able now to use sine rule to find the areas;find attached...

1613385094504.png


what i was missing in this question was how to come up with the linear scale dimensions,
##0.75##, ##0.5## and ##0.25## for the 1st, 2nd and 3rd triangles respectively...
 
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  • #23
chwala said:
linear scale dimensions for the 1st, 2nd and 3rd triangles respectively...
But once you have those, it is obvious that the answer is
$$6 \times \frac{3}{4} \times (1 + (\frac{2}{3})^2 + (\frac{1}{3})^2 )$$ :biggrin:
 
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  • #24
1613495993292.png
 
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  • #25
Does the spare triangle make any sense ?
 
  • #26
Zgort said:
While it is true that the green area above the diagonal is equal to the white area below the diagonal, there is no reasoning given to show that the green area above the diagonal equals the white area above.
In other words, the is no reasoning given to show that half of each half is green and half of each half is white. In fact that's not true.
 
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What is the definition of area?

The area is the measurement of the surface enclosed by a shape or figure.

How do you find the area of a rectangle?

The area of a rectangle can be found by multiplying the length by the width. The formula for area of a rectangle is A = l x w, where A is the area, l is the length, and w is the width.

What is the formula for finding the area of a circle?

The formula for finding the area of a circle is A = πr^2, where A is the area, π is the mathematical constant pi, and r is the radius of the circle.

How do you find the area of a triangle?

The area of a triangle can be found by multiplying the base by the height and then dividing by 2. The formula for area of a triangle is A = (b x h) / 2, where A is the area, b is the base, and h is the height.

What is the process for finding the area of a shaded region?

The process for finding the area of a shaded region depends on the shape of the region. For regular shapes like squares, rectangles, and circles, you can use the appropriate formula. For irregular shapes, you may need to break the shape into smaller, simpler shapes and find the area of each one before adding them together to get the total area of the shaded region.

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