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(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}). 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

x_{1} + x_{2} + x_{2} + x_{3} + x_{1} + x_{3} = 6 + 8 + 10

⇒ 2(x_{1} + x_{2} + x_{3}) = 24

⇒ x_{1} + x_{2} + x_{3} =12 …(vii)

From (i) and (vii), we get

x_{3} = 12 – 6 = 6

From (iii) and (vii), we get

x_{1} = 12 – 8 = 4

From (v) and (vii), we get

x_{2} = 12 – 10 = 2

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

y_{1} + y_{2} + y_{2} + y_{3} + y_{1} + y_{3} = 8 + 12 + 14

⇒ 2(y_{1} + y_{2} + y_{3}) = 34

⇒ y_{1} + y_{2} + y_{3} = 17 …(viii)

From (ii) and (viii), we get

y_{3} = 17 – 8 = 9

From (iv) and (vii), we get

y_{1} = 17 – 12 = 5

From (vi) and (vii), we get

y_{2} = 17 – 14 = 3

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

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