Question

A (4, 2), B(6, 5) and C (1, 4) are the vertices of ΔABC.

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

(ii) Find the coordinates of point P and AD such that AP : PD = 2 : 1.

(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1

(iv) What do you observe?

Answer 1

We have triangle in which the co-ordinates of the vertices are A (4, 2); B (6, 5) and C (1, 4)

(i)It is given that median from vertex A meets BC at D. So, D is the mid-point of side BC.

In general to find the mid-point of two pointsand we use section formula as,

Therefore mid-point D of side BC can be written as,

Now equate the individual terms to get,

So co-ordinates of D is

(ii)We have to find the co-ordinates of a point P which divides AD in the ratio 2: 1 internally.

Now according to the section formula if any point P divides a line segment joining andin the ratio m: n internally than,

P divides AD in the ratio 2: 1. So,

(iii)We need to find the mid-point of sides AB and AC. Let the mid-points be F and E for the sides AB and AC respectively.

Therefore mid-point F of side AB can be written as,

So co-ordinates of F is

Similarly mid-point E of side AC can be written as,

So co-ordinates of E is

Q divides BE in the ratio 2: 1. So,

Similarly, R divides CF in the ratio 2: 1. So,

(iv)We observe that that the point P, Q and R coincides with the centroid. This also shows that centroid divides the median in the ratio 2: 1.