Question

What is the perimeter of an equilateral triangle whose height is 2(radical 3)?

Answer 1

Socratic Formatting for radical is : hashsymbol sqrt(3) hashsymbol giving: `sqrt(3)`. Look at https://socratic.org/help/symbols.

Perimeter = 4

Explanation:

Let each triangle side be of length `x`

Let height be `h`

Then, by using Pythagoras

`h^2+(x/2)^2=x^2`

subtract `(x/2)^2` from both sides

`h^2=x^2-(x/2)^2`

`h^2=(4x^2)/4-x^2/4`

`h^2=3/4x^2`

Multiply both sides by `4/3`

`4/3 h^2=x^2`

Square root both sides

`x=(2h)/sqrt(3)`

Mathematicians do not like the denominator to be a radical

Multiply the right by 1 but in the form of `1=sqrt(3)/(sqrt(3)`

`x=(2hsqrt(3))/3`

But `h=2sqrt(3)` so by substitution for `h`

`x=(2(2sqrt(3))sqrt(3))/3`

`x=12/3=4`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Triangle has 3 sides and each side is 4

Perimeter is `3xx4=12`