What is the perimeter of an equilateral triangle if the length of an altitude is 5/sqrt3?

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Answer 1 · 2016-03-13T17:37:13

Perimeter is 10

$color(red)("Using ratios is a very powerful tool!")$

Tony B

Let the height of the standardises triangle by $h$
Let the side length of the triangle in the question be $x$

The by ratio of side lengths we have:

$color(blue)(("height of target triangle")/("height of standard triangle")= ("side of target triangle")/("side of standard triangle"))$

$(5/sqrt(3))/h = x/2$

$5/sqrt(3)xx1/h=x/2$

But $h= sqrt(3)$ giving

$5/sqrt(3)xx1/sqrt(3) =x/2$

$x=(2xx5)/3$

But this is the length for only one side. There are three sides so:

Perimeter = $(3xx2xx5)/3 = 3/3 xx2xx5=10$