# NCERT Class 11 Mathematics Solutions: Chapter 12 – Introduction to Three Dimensional Geometry Miscellaneous Exercise Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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1. Three vertices of a parallelogram are . Find the coordinates of the fourth vertex.

The three vertices of a parallelogram are given as .

Consider the coordinates of the fourth vertex be .

We know that the diagonals of a parallelogram bisect each other.

So, in parallelogram bisect each other.

So, the coordinates of the fourth vertex are .

2. Find the lengths of the medians of the triangle with vertices

Consider be the medians of the given triangle .

Since is the median, is the mid-point of .

Coordinates of point

Since is the median, is the mid-point of

Coordinates of point

Since is the median, is the-point of .

Coordinates of point

Length of

So, the length of the medians of are .

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