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

Question

5430 views around the world

You can reuse this answer
Creative Commons License

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#