In D ABC, G (-4, -7) is the centroid. If A (-14, -19) and B(3, 5) then find the co-ordinates of C.
In Δ ABC, G (-4, -7) is the centroid. If A (-14, -19) and B(3, 5) then find the co-ordinates of C.
Answer:-
Let the coordinates of point C be (x, y).
Centroid = G (–4, –7)
[- 4 = left( frac{- 14 + 3 + x}{3} right)]
[ Rightarrow – 12 = x – 11]
[ Rightarrow x = – 1]
[text { Also }, ]
[ – 7 = left( frac{- 19 + 5 + y}{3} right)]
[ Rightarrow – 21 = – 14 + y]
[ Rightarrow y = – 7]
Thus, point C will be
[left( – 1, – 7 right)].
Explanation:-
We know that the centroid of a triangle is the point of intersection of its medians.
Let’s first find the co-ordinates of point C by using the midpoint formula:
Midpoint of AB = ( (x1 + x2)/2 , (y1 + y2)/2 ) Midpoint of AB = ( (-14 + 3)/2 , (-19 + 5)/2 ) Midpoint of AB = ( -5.5 , -7 )
We know that the centroid divides each median in a 2:1 ratio. So, we can use this to find the co-ordinates of point C as follows:
GC = 2/3 * GA GC = 2/3 * ( (-14/3, -19/3) – (-4, -7) ) GC = 2/3 * (-14/3 + 4, -19/3 + 7) GC = 2/3 * (-2/3, 2/3) GC = (-4/9, 2/9)
Similarly, we can find BC by using the midpoint formula:
Midpoint of AC = ( (x1 + x2)/2 , (y1 + y2)/2 ) Midpoint of AC = ( (-14 + x)/2 , (-19 + y)/2 ) Midpoint of BC = ( (3 + x)/2 , (5 + y)/2 )
Since G is the centroid, we know that:
G = ( (x1 + x2 + x3)/3 , (y1 + y2 + y3)/3 ) (-4, -7) = ( (-14 + x + 3)/3 , (-19 + y + 5)/3 ) (-12, -12) = ( -11 + x/3 , -14 + y/3 ) x/3 = -1 and y/3 = 2 x = -3 and y = 6
So, the co-ordinates of point C are (-3, 6).
Chapter 5. Co-ordinate Geometry – Practice Set 5.2 (Page 115)