Question

Distance between centroid and vertex of equilateral triangle

Answer 1

the centroid is always located in the interior of the triangle. The centroid is located 2/3 of the distance from the vertex along the segment that connects the vertex to the midpoint of the opposite side.

Answer 2

Answer:

Distance Between Centroid & vertex of equilateral triangle is  $\frac{x}{\sqrt{3}}$

Step-by-step explanation:

Given: Centroid and vertex of Equilateral triangle

To find: Distance between centroid and vertex.

Centroid is the point of intersection of all 3 medians of triangle.

Also, In equilateral triangle median and Altitude are same.

A centroid divide a median in 2 : 1. ratio.

Let x be the length of equilateral triangle ABC,

CB = $\frac{1}{2}\times AB=\frac{x}{2}$

BD is median as well as altitude.

Using Pythagoras theorem in ΔABD

AB² = AD² + BD²

$x^2=(\frac{x}{2})^2+BD^2$

$BD^2=x^2-\frac{x^2}{4}$

$BD^2=\frac{3x^2}{4}$

$BD=\frac{\sqrt{3}x}{2}$

⇒ Length of median = $\frac{\sqrt{3}x}{2}$

Total part of ratio = 1 + 2 = 3

⇒ Distance between Centroid & vertex of equilateral triangle = $\frac{2}{3}\times\frac{\sqrt{3}x}{2}\:=\:\frac{x}{\sqrt{3}}$

Therefore, Distance Between Centroid & vertex of equilateral triangle is  $\frac{x}{\sqrt{3}}$