Question

Let P be a point in an equilateral triangle with each side of length '1'. Let `h_1,h_2,h_3` be the length of perpendicular distance from 'P' to the 3 sides of the triangle. What is the possible value of `h_1+h_2+h_3`?

Answer 1

Area of the equilateral triangle having length of each side 1 is given by `sqrt3/4*1^2=sqrt3/4`squnit

Again it's area= sum of areas of three triangles obtained by joining P with the three vertices.

`1/2*1*h_1+1/2*1*h_2+1/2*1*h_3`

So`1/2(h_1+h_2+h_3)=sqrt3/4`

`=>h_1+h_2+h_3=sqrt3/2`

Answer 2

See explanation.

Explanation:

For any equilateral #triangle ABC of side 'a',

the area of the triangle`= 1/2(sqrt3/2 a)( a ) = sqrt3/4a^2`

= sum of the areas of `triangle`s `PBC, PCA and PAB`

`= 1/2 ( h_1 + h_2 + h_3 ) a`, and so,

`( h_1 + h_2 + h_3 ) = (sqrt3/2) a`

This property of equilateral triangles is used in making

Triangular Graphs, for studying miscibility of liquids.

https://www.google.co.in/search?q=Triangular+Graphs&rlz=1C1GIGM_enIN777IN777&oq=Triangular+Graphs&aqs=chrome