# CBSE Class 10-Mathematics: Chapter – 7 Coordinate Geometry Part 27 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 23**:

Let and be the vertices of

(i) The median from A meets BC at D. Find the coordinates of the point D.

(ii) Find the coordinates of the point P on AD such that .

(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that .

(iv) What do you observe?

(Note: The point which is common to all the three medians is called centroid and this point divides each median in the ratio )

(v) If A are the vertices of , find the coordinates of the centroid of the triangle.

**Answer**:

Let and be the vertices of

(i) Since AD is the median of .

D is the mid-point of BC.

Its coordinates are

(ii) Since P divides AD in the ratio

Its coordinates are

(iii) Since BE is the median of .

E is the mid-point of AD.

Its coordinates are

Since Q divides BE in the ratio .

Its coordinates are

Since CF is the median of .

F is the mid-point of AB.

Its coordinates are

Since R divides CF in the ratio .

Its coordinates are

(iv) We observe that the points P, Q and R coincide, i.e.. , the medians AD, BE and CF are concurrent at the point . This point is known as the centroid of the triangle.

(v) According to the question, D, E, and F are the mid-points of BC, CA, and AB, respectively.

Coordinates of a point dividing AD in the ratio are

The coordinates of E are

The coordinates of a point dividing BE in the ratio are

Similarly, the coordinates of a point dividing CF in the ratio are

Thus, the point is common to AD, BE and CF and divides them in the ratio .

The median of a triangle are concurrent, and the coordinates of the centroid are