The mid-points of sides AB, BC, CA of triangle ABC are D(2,1), E(1,0), F(-1,3), respectively. Find the coordinates of the vertices of triangle ABC. plz explain step by step(plz answer fast)

Asked by Rutuja Mehta | 23rd Oct, 2013, 08:33: AM

Expert Answer:

The triangle is ABC. 
The mid-points are D (2,1), E (1,0) and F(-1,3) for the sides AB,BC and CA respectively. 
Assuming the coordinates of the points A, B and C as (x1, y1), (x2, y2) and (x3, y3): 

The mid point's coordinates are given by the arithmetic mean of the coordinates of end-points. 
For D it will be: D = [(x1+ x2) / 2] = 2 and similarly for E and F. 
So, (x1+ x2) = 2 x 2 = 4  ---(1) 
Similarly, (x2+ x3) = 2 x 1 = 2   ---(2)
And (x3+ x1) = 2 x (-1) = -2    ---(3) 
 
Adding the equations 1, 2 and  3 we get: 2 (x1+ x2 + x3) = 4 <==> (x1+ x2 + x3) = 2 ---(4)
Subtract equations 1, 2, 3 from 4 to get values of x1, x2 and x3
And the values of  x1, xand x3 are 0, 4 and -2. 
 
Similarly, we will perform the operation for y-coordinates: 
D: [(y1+y2)/2] = 1
Hence, y1+y2 = 2 x 1 = 2  ---(5)
Similarly, y+ y3 = 2 x 0 = 0   ---(6)
And y3 + y1 = 2 x 3 = 6    ---(7)
Adding equation 5, 6 and 7 we get: 2(y1 + y2 + y3) = 8 <==> (y1 + y2 + y3) = 4   ---(8)
 
Subtract the equations 5, 6 and 7 from 8 to get values of y1, y2 and y3
So, the values of y1, y2 and y3 are 4, -2 and 2 respectively. 
 
Hence, (x1,y1) = (0, 4)
(x2, y2) = (4, -2)
and (x3, y3) = (-2, 2)  

Answered by  | 23rd Oct, 2013, 10:00: AM

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