<s>[INST] Find the distance of the centroid of the triangle ABC from the origin? A sphere of radius $r$ passes through the origin and the other points where it meets the coordinate axes are A, B and C. Find the distance of the centroid of the triangle ABC from the origin (in terms of $r$).
i was taking the coordinate of the centroid ($\frac{x+y+z}{3}$,  $\frac{X+Y+Z}{3}$).   but   i  don't know the distance  of the centroid  of triangle ABC  from the origin. From my point of view  i was using the distance formula  from the origin,, but i didn't get it
if anybody help me me  i would be very thankful to him
thank you
 [/INST] If the sphere intersects the three axes at the points A, B and C, it means that the sphere passes through these three points:
$(A,0,0)$; $(0,B,0)$; $(0,0,C)$
i.e., the sphere intersects x-axis at $(A,0,0)$ and y-axis at $(0,B,0)$ and z-axis at $(0,0,C)$
Now, coordinates of the centroid of triangle formed by these three co-ordinates will be:
$(\frac{A+0+0}{3},\frac{0+B+0}{3},\frac{0+0+C}{3})$ = $(\frac{A}{3},\frac{B}{3},\frac{C}{3})$
To find distance of this centroid from origin $(0,0,0)$ we simply use the two point distance formula in 3d:
Distance between $(\frac{A}{3},\frac{B}{3},\frac{C}{3})$ and $(0,0,0)$ = 
$\sqrt{(\frac{A}{3}-0)^2 + (\frac{B}{3}-0)^2 + (\frac{C}{3}-0)^2}$ = $\frac{\sqrt{A^2 + B^2 + C^2}}{3}$

In our case, the radius of sphere is given as $r$.
Let's assume the centre of sphere to be at $(x_0,y_0,z_0)$. Then the equation of the sphere can be written as:
$(x-x_0)^2+(y-y_0)^2+(z-z_0)^2 = r^2$
Since it's also given that the sphere passes through the origin $(0,0,0)$, this point must satisfy the above equation of sphere.
Or, $(0-x_0)^2+(0-y_0)^2+(0-z_0)^2 = r^2$
Or, $x_0^2 + y_0^2 + z_0^2 = r^2$    --- $(1)$
Now, the sphere also passes through the point $(A,0,0)$. Substituting this point in the equation of sphere, we have:
$(A-x_0)^2+(0-y_0)^2+(0-z_0)^2 = r^2$
Or, $(A-x_0)^2+y_0^2+z_0^2 = r^2$
From $(1)$, let's also substitute $r^2$ with $x_0^2 + y_0^2 + z_0^2$
$(A-x_0)^2+y_0^2+z_0^2 = x_0^2 + y_0^2 + z_0^2$
Or, $(A-x_0)^2 = x_0^2$
Or, $A-x_0 = x_0$
Thus we have $A = 2x_0$.
Similarly by substituting $(0,B,0)$ and $(0,0,C)$ in the equation of sphere we get $B = 2y_0$ and $C= 2z_0$.
Hence, the three points A, B and C are $(2x_0,0,0)$, $(0,2y_0,0)$ and $(0,0,2z_0)$ respectively.
Thus, by using the formula we arrived at in section 1, we can say the distance of centroid of triangle ABC from origin = 
$\frac{\sqrt{A^2 + B^2 + C^2}}{3}$ = $\frac{\sqrt{(2x_0)^2 + (2y_0)^2 + (2z_0)^2}}{3}$ = $\frac{2\sqrt{x_0^2 + y_0^2 + z_0^2}}{3}$
But from equation $(1)$ we know that $x_0^2 + y_0^2 + z_0^2 = r^2$.
Substituting this we have the required distance as $\frac{2\sqrt{r^2}}{3} = \frac{2r}{3}$
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