We know, distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is \(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\).
So, distance between two points (5, 6, 7) and (2, 6, 3) will be \(\sqrt{(5-2)^2+(6-6)^2+(7-3)^2}\) = \(\sqrt{(3)^2+(4)^2}\) = 5 units.

We know, distance of a point from X-Z plane is equal to value of its y-coordinate.
So, distance of point (2, 3, 5) from X-Z plane is 3 units.

If coordinates of vertices of a triangle are (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) coordinates of centroid of triangle are ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
So, coordinates of centroid of given triangle are ((7+5+9)/3, (6+4+5)/3, (4+6+8)/3) = (7, 5, 3).

We know, distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is \(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\).
Distance AB = \(\sqrt{(7-6)^2+(0+1)^2+(10-6)^2} = \sqrt{18}\)
Distance BC = \(\sqrt{(6-9)^2+(-1+4)^2+(6-6)^2} = \sqrt{18}\)
Distance AC = \(\sqrt{(7-9)^2+(0+4)^2+(10-6)^2} = \sqrt{36}\)
Since AB = BC and AB²+BC² = AC² so, it forms right angled isosceles triangle.

The coordinates of a point dividing line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in ratio m : n is \((\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n})\).
Let ratio be k : 1.
So, z-coordinate of point will be (k*6+1*3)/(k+1).
We know, for X-Y plane, z coordinate is zero.
(6k+1*3)/(k+1) = 0 => k=-1/2

Since point with only y coordinate positive i.e. of form (-, +, -) lie in 6^th octant.
So, (-1, 5, -7) lies in 6^th octant.

In 3d coordinate system, a point with x and y coordinate zero and z-coordinate having non-zero value must lie on z-axis. So, (0, 0, 3) lie on z-axis.

We know, midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) is ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).
Midpoint of line PR is (5, 4, 6).
Length of median through Q is distance between midpoint of PR and Q i.e. \(\sqrt{(5-2)^2+(4-5)^2+(6-5)^2} = \sqrt{(3)^2+(-1)^2+(1)^2} = \sqrt{11}\).

The coordinates of a point dividing line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in ratio m: n is \((\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n})\).
Let ratio be k : 1.So, coordinates of a point dividing line segment joining (1, 2, 3) and (4, 5, 6) internally in ratio k: 1 is \((\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1},\frac{k*6+1*3}{k+1})\)
=> \((\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1},\frac{k*6+1*3}{k+1})\) is same as (3, 4, 5).
=> (4k+1)/(k+1) = 3
=> 4k+1 = 3k+3
=> k = 2
So, ratio is 2:1.

3-D geometry can have 3 coordinates each with positive or negative value.
Total octants = 2*2*2 = 8.