Chapter 12.2, Problem 2C

To determine

To find: The value of $x+y+z$ and explain the observation from the triangle.

Answer to Problem 2C

It can be noticed that the sum of the distances from any interior point to the sides of the equilateral is equal to the length of triangle’s altitude.

Explanation of Solution

Given information: An equilateral triangle such that x , y and z are the perpendicular distances from any point inside the triangle to each side. The altitude of the triangle is h .

Calculation:

Choose another point Q inside the triangle and x , y and z are the perpendicular distances from this point to each side. Let, the length of each side of equilateral triangle be a .

  

Figure (1)

Draw a line from $P$ to each of $A$ , $B$ , and $C$ forming three triangles $PAB$ , $PBC$ , and $PCA$ .

The formula to find the area of triangle is:

  $\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$

So,

  $\begin{array}{l}\text{Area}\text{of}\Delta PAC=\frac{a.z}{2}\\ \text{Area}\text{of}\Delta PAB=\frac{a.x}{2}\\ \text{Area}\text{of}\Delta PCB=\frac{a.y}{2}\end{array}$

The area of $\Delta ABC$ can be calculated as:

  $\begin{array}{c}\Delta ABC=\Delta PAC+\Delta PAB+\Delta PCB\\ \frac{a.h}{2}=\frac{a.z}{2}+\frac{a.x}{2}+\frac{a.y}{2}\\ x+y+z=h\end{array}$

It can be noticed that the sum of the distances from any interior point to the sides of the equilateral $\Delta ABC$ is equal to the length of triangle’s altitude.

It happens because every center an equilateral triangle coincides with its centroid, this implies that an equilateral triangle is the only triangle with no Euler line connecting some of the centers.

Therefore, it can be noticed that the sum of the distances from any interior point to the sides of the equilateral is equal to the length of triangle’s altitude.