# NCERT Class 11-Math's: Chapter –12 Introduction to Three Dimensional Geometry Part 9

Question 18:

Show that the three points , and are collinear and find the ratio in which C divides AB.

Given; Three points , and

; Points A, B and C are collinear.

∴ From the lengths of AB, BC and AC we can say that C divides AB in the ratio 2:1 externally.

Question 19:

The mid-point of the sides of a triangle are and . Find its vertices. Also find the centriod of the triangle.

Given; The mid-point of the sides of a triangle are and .

Let he vertices be , and respectively.

,

.

,

,

.

,

,

.

are the required vertices.

Centroid of a triangle is given by the average of the coordinates of its vertices or midpoint of sides.

Centroid is

Question 20:

Prove that the points , and are collinear. Find the ratio in which the first point divides the join of the other two.

Given; Three points , and

; ∴ Points A, B and C are collinear.

∴ From the lengths of AB, BC and AC we can say that the first point divides the join of the other two in the ratio externally.

Question 21:

What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?

Given; Cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin.

The Coordinates of the vertices are;

,

,

,

,

,

,

,

.

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