How do you find the perimeter of an equilateral with an altitude of #6sqrt3# cm?

Question

Evelyn Arboleda · Accepted Answer

Apply the Pythagorean theorem to solve for the side length and find that the perimeter is #36"cm"#

Drawing it out, we obtain the following picture:

Our goal is to find the perimeter, that is, #s+s+s=3s#

Looking at the right triangle with the sides marked in the picture, we can apply the Pythagorean theorem to get

#(s/2)^2 + h^2 = s^2#

#=> s^2/4 + h^2 = s^2#

#=> h^2 = 3/4s^2#

#=> s^2 = (4h^2)/3#

#=> s = sqrt((4h^2)/3)#

Substituting in our value for #h#, we obtain

#s = sqrt((4*(6sqrt(3))^2)/3)#

#=sqrt((4*108)/3)#

#=sqrt(144)#

#=12#

Thus #3s = 3*12 = 36#

So the perimeter is #36"cm"#

Allison Allen · Answer

undefined To find the perimeter of an equilateral triangle with an altitude of $ 6\sqrt{3} $ cm, you can use the formula:

$$ \text{Perimeter} = 3 \times \text{side length} $$

Given that the altitude is $ 6\sqrt{3} $ cm, you can find the side length using the formula for the altitude of an equilateral triangle:

$$ \text{altitude} = \frac{\sqrt{3}}{2} \times \text{side length} $$

Solve this equation for the side length:

$$ 6\sqrt{3} = \frac{\sqrt{3}}{2} \times \text{side length} $$

$$ \text{side length} = \frac{6\sqrt{3} \times 2}{\sqrt{3}} $$

$$ \text{side length} = 12 $$

Now that you have the side length, you can find the perimeter:

$$ \text{Perimeter} = 3 \times 12 $$

$$ \text{Perimeter} = 36 \text{ cm} $$

So, the perimeter of the equilateral triangle is $ 36 $ cm.