If the mid points of sides of ΔABC are (2,3),(3,7) and (−2,1), then the possible coordinates of its vertices are
A
(−1,5)
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B
(−3,−3)
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C
(7,9)
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D
(5,−1)
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Solution
The correct options are A(−1,5) B(−3,−3) C(7,9) Let A(x1,y1),B(x2,y2),C(x3,y3) M1(2,3) is midpoint of side BC M2(3,7) is midpoint of side AC M3(−2,1) is midpoint of side AB
Adding equation (1),(2) and (3), 2(x1+x2+x3)=6⇒x1+x2+x3=3⋯(4) Using equation (1) and (4), x3=7 Using equation (2) and (4), x1=−1 Using equation (3) and (4), x2=−3
Similarly, for y−coordinates y1+y2=2⋯(5)y2+y3=6⋯(6)y3+y1=14⋯(7)
Adding equation (5),(6) and (7) y1+y2+y3=11⋯(8) Using equation (5) and (8), y3=9 Using equation (6) and (8), y1=5 Using equation (7) and (8), y2=−3
Hence, the coordinates of the triangle vertex are A=(−1,5),B=(−3,−3),C=(7,9)
Alternate Solution :
Let A(x1,y1),B(x2,y2),C(x3,y3) M1(2,3) is midpoint of side BC M2(3,7) is midpoint of side AC M3(−2,1) is midpoint of side AB
We know that, BM1M2M3 is a parallelogram. And its diagonal bisect each other, (x2+32,y2+72)=(2−22,3+12)⇒x2=−3,y2=−3∴B=(−3,−3)
Similarly, CM1M3M2 is a parallelogram. (x3−22,y3+12)=(2+32,3+72)⇒x3=7,y3=9∴C=(7,9)
Similarly, AM3M1M2 is a parallelogram. (x1+22,y1+32)=(3−22,7+12)⇒x1=−1,y1=5∴A=(−1,5)