Q. 8 C4.0( 6 Votes )

The mid-points of the sides of a triangle are (3,4),(4,6) and (5,7). Find the coordinates of the vertices of the triangle.

Answer :


Consider a ΔABC with A(x1, y1), B(x2, y2) and C(x3, y3). If P(3, 4), Q(4, 6) and R(5, 7) are the midpoints of AB, BC, and CA. Then,


…(i)


…(ii)


…(iii)


…(iv)


…(v)


…(vi)


Adding (i), (iii) and (v), we get


x1 + x2 + x2 + x3 + x1 + x3 = 6 + 8 + 10


2(x1 + x2 + x3) = 24


x1 + x2 + x3 =12 …(vii)


From (i) and (vii), we get


x3 = 12 – 6 = 6


From (iii) and (vii), we get


x1 = 12 – 8 = 4


From (v) and (vii), we get


x2 = 12 – 10 = 2


Now adding (ii), (iv) and (vi), we get


y1 + y2 + y2 + y3 + y1 + y3 = 8 + 12 + 14


2(y1 + y2 + y3) = 34


y1 + y2 + y3 = 17 …(viii)


From (ii) and (viii), we get


y3 = 17 – 8 = 9


From (iv) and (vii), we get


y1 = 17 – 12 = 5


From (vi) and (vii), we get


y2 = 17 – 14 = 3


Hence, the vertices of ΔABC are A(4, 5), B(2, 3) and C(6, 9)


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