Question

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.

Answer 1

Let three vertices be A(1, -2), B(3, 6) and C(5, 10) and fourth vertex be D(x, y) 

It is given that quadrilateral joining these four vertices is parallelogram, ie □ABCD is parallelogram. 

We know that diagonals of parallelogram bisect each other, ie midpoint of the diagonals coincide.

Let E(x_m , y_m) be the midpoint of diagonals AC and BD. 

By midpoint formula,

x = \(\frac{x_1+x_2}2\), y = \(\frac{y_1+y_2}2\)

For diagonal AC,

x_m = \(\frac{1+5}2\), y_m = \(\frac{-2 +10}2\)

∴ x_m = \(\frac{6}2\) , y_m = \(\frac{8}2\)

∴ E(x_m , y_m) ≡ (3, 4) 

For diagonal BD,

 3 = \(\frac{3+x}2\), 4 = \(\frac{6 +y}2\)

∴ x = 6 – 3 , y = 8 – 6 

∴ x = 3 and y = 2 

Hence, 

our fourth vertex is D(3 , 2)