Q2. - Geometry

Question

Now choose another point on or inside the triangle and find$x+y+z$ . What do you notice? Why does this happen?

Step-by-Step Solution

Answer

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.

Step 1. 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.

Step 2. Draw the triangle.

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.

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

Step 3. Explanation.

The formula to find the area of triangle is:

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

 

So,

$\begin{array}{c}\text{Area\hspace{0.17em}of\hspace{0.17em}}\Delta PAC=\frac{a.z}{2}\\ \text{Area\hspace{0.17em}of\hspace{0.17em}}\Delta PAB=\frac{a.x}{2}\\ \text{Area\hspace{0.17em}of\hspace{0.17em}}\Delta PCB=\frac{a.y}{2}\end{array}$

Step 4. Determine the value.

So, the area of 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 centres.

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.