# Find the lengths of the medians of the triangle with vertices A(0, 0, 6), B (0, 4, 0) and C (6, 0, 0)

**Solution:**

Let AD, BE and CF be the medians of the given triangle, where D, E, and F are the midpoints of the sides BC, AC, and AB respectively.

Since, AD is the median, D is the mid-point of BC.

Coordinates of point D = [(0 + 6)/2, (4 + 0)/2, (0 + 0)/2] = (3, 2, 0)

Now, using the 3d distance formula,

AD = √(0 - 3)² + (0 - 2)² + (6 - 0)²

= √9 + 4 + 36

= √49

= 7

Since, BE is the median, E is the mid-point of AC.

Coordinates of point E = [(0 + 6)/2, (0 + 0)/2, (0 + 6)/2] = (3, 0, 3)

Now, using the 3d distance formula,

BE = √(3 - 0)² + (0 - 4)² + (3 - 0)²

= √9 + 16 + 9

= √34

Since CF is the median, F is the mid-point of AB.

Coordinates of point F = [(0 + 0)/2, (0 + 4)/2, (0 + 6)/2] = (0, 2, 3)

Now, using the 3d distance formula,

CF = √(6 - 0)² + (0 - 2)² + (0 - 3)²

= √36 + 4 + 9

= √49

= 7

Thus, the lengths of the medians of triangle ABC are 7, √34 and 7

NCERT Solutions Class 11 Maths Chapter 12 Exercise ME Question 2

## Find the lengths of the medians of the triangle with vertices A(0, 0, 6), B (0, 4, 0) and C (6, 0, 0)

**Summary:**

The lengths of the medians of the triangle with vertices A(0, 0, 6), B (0, 4, 0) and C (6, 0, 0) are 7, √34 and 7