# 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?

Jul 31, 2018

Area of the equilateral triangle having length of each side 1 is given by $\frac{\sqrt{3}}{4} \cdot {1}^{2} = \frac{\sqrt{3}}{4}$squnit

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

$\frac{1}{2} \cdot 1 \cdot {h}_{1} + \frac{1}{2} \cdot 1 \cdot {h}_{2} + \frac{1}{2} \cdot 1 \cdot {h}_{3}$

So$\frac{1}{2} \left({h}_{1} + {h}_{2} + {h}_{3}\right) = \frac{\sqrt{3}}{4}$

$\implies {h}_{1} + {h}_{2} + {h}_{3} = \frac{\sqrt{3}}{2}$

Jul 31, 2018

See explanation.

#### Explanation:

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

the area of the triangle$= \frac{1}{2} \left(\frac{\sqrt{3}}{2} a\right) \left(a\right) = \frac{\sqrt{3}}{4} {a}^{2}$

= sum of the areas of $\triangle$s $P B C , P C A \mathmr{and} P A B$

$= \frac{1}{2} \left({h}_{1} + {h}_{2} + {h}_{3}\right) a$, and so,

$\left({h}_{1} + {h}_{2} + {h}_{3}\right) = \left(\frac{\sqrt{3}}{2}\right) a$

This property of equilateral triangles is used in making

Triangular Graphs, for studying miscibility of liquids.