Equilateral triangle, perimeter and area? idk what to call this

toidinami

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Oct 16, 2018
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An equilateral triangle has the perimeter of 24.

Show that the area of the triangle is 16*(sqrt of 3)


Some have said just to confirm it, but is there any correlation or method to come to the conclusion that the area is equal to 16*(sqrt of 3) from the given information?
 
Usually when a math problem asks you to "show [such and such]", they want you to apply some known facts/definitions/formulas in order to reach the desired conclusion. So let's do that now. A great place to start is: What is the definition of an equilateral triangle? From this definition and given that the perimeter of the equilateral triangle is 24, what must be the side lengths of said triangle? What is the formula for the area of a triangle? Which side of the triangle is the base, and what is its length? If you draw an imaginary line splitting the equilateral triangle into two right triangles, can you use the Pythagorean Theorem to find the height of the equilateral triangle? Finally, plugging these new-found values into the formula for the area, what do you get?
 
To extend Denis' hint, if you draw a line from one vertex of an equilateral triangle perpendicular to the base, that line also bisects the side, dividing the equilateral triangle into two congruent triangles. Since the equilateral triangle has perimeter 24, each side of the triangle has length 24/3= 8. The two congruent triangles have hypotenuse of length 8 and one leg of length 8/2= 4. What is the length of the other leg?
 
Yeah I had figured out that much. But figured out where i went wrong.
I came to the part where i figured out that the height = sqrt(48), but then i always walked past this and calculated it to a number when i was supposed to use this to get to 16*sqrt(3)

the second reason i got stuck was because i just didnt see or think of the possibility that i could "factor out a perfect square" like i did in step 3

Final solution:

A=(h*s)/2

=(sqrt(48) * 8) / 2

=(sqrt(4^2)*sqrt(3) *8) / 2

=(4 * 8 * sqrt(3)) / 2

=(32 * sqrt(3)) /2

=16 * sqrt(3)

Thanks for your help!
 
An equilateral triangle has the perimeter of 24.

Show that the area of the triangle is 16*(sqrt of 3)


Some have said just to confirm it, but is there any correlation or method to come to the conclusion that the area is equal to 16*(sqrt of 3) from the given information?

You could use the Heron's formula, after finding the side of the triangle (24/3=8). So,
p = P/2 = 12
A = sqrt(p * (p-a) * (p-b) * (p-c)) = sqrt(12 * 43) = 16 * sqrt(3)
 
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