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Solution


Q.

If the coordinates of the midpoints of the sides of the triangle are (1, 2), (0, −1) and (2, −1), which of the following coordinates are the vertices of the triangle

(i) (1, −4)  (ii) (−1, 2)  (iii) (3, 2)  (iv) (2, 3)  (v) (−4, 1)

Choose the correct option from the following:
1  (i), (ii) and (iii)
 
2 (ii), (iii) and (iv)  
3 (iii), (iv) and (v)  
4 (i), (iii) and (iv)  

Ans.

(1)

Let A = (x₁, y₁), B = (x₂, y₂), C = (x₃, y₃)be the vertices of ΔABC.Let the midpoints be:D(1, 2) on BC, E(0, −1) on CA, F(2, −1) on ABSince D is the midpoint of BC:x2+x32=1 and y2+y32=2⇒x2+x3=2, y2+y3=4...(i)Since E is the midpoint of AC:x1+x32=0andy1+y32=-1⇒x1+x3=0, y1+y3=-2...(ii)Since F is the midpoint of AB:x1+x22=2andy1+y22=-1⇒x1+x2=4, y1+y2=-2...(iii)Adding (i), (ii) and (iii):(x2+x3)+(x1+x3)+(x1+x2)=2+0+4⇒x1+x2+x3=3Similarly for y-values:(y2+y3)+(y1+y3)+(y1+y2)=4-2-2⇒y1+y2+y3=0...(iv)Finding coordinates of A