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

Apr 28, 2016

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

${h}^{2} + {\left(\frac{x}{2}\right)}^{2} = {x}^{2}$

subtract ${\left(\frac{x}{2}\right)}^{2}$ from both sides

${h}^{2} = {x}^{2} - {\left(\frac{x}{2}\right)}^{2}$

${h}^{2} = \frac{4 {x}^{2}}{4} - {x}^{2} / 4$

${h}^{2} = \frac{3}{4} {x}^{2}$

Multiply both sides by $\frac{4}{3}$

$\frac{4}{3} {h}^{2} = {x}^{2}$

Square root both sides

$x = \frac{2 h}{\sqrt{3}}$

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)

$x = \frac{2 h \sqrt{3}}{3}$

But $h = 2 \sqrt{3}$ so by substitution for $h$

$x = \frac{2 \left(2 \sqrt{3}\right) \sqrt{3}}{3}$

$x = \frac{12}{3} = 4$

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Triangle has 3 sides and each side is 4

Perimeter is $3 \times 4 = 12$