Question

A quadrilateral has vertices (4, 1), (1, 7), (– 6, 0) and (– 1, – 9). Show that the mid – points of the sides of this quadrilateral form a parallelogram.

Answer 1

Given, A quadrilateral has vertices (4, 1), (1, 7), (– 6, 0) and (– 1, – 9).

To Prove: Mid – Points of the quadrilateral form a parallelogram. 

The formula used: Mid point formula = \(\Big[\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\Big]\)

Explanation: Let ABCD is a quadrilateral 

E is the midpoint of AB 

F is the midpoint of BC 

G is the midpoint of CD 

H is the midpoint of AD 

Now, Find the Coordinates of E, F,G and H using midpoint Formula

Now, EFGH is a parallelogram if the diagonals EG and FH have the same mid – pointNow, EFGH is a parallelogram if the diagonals EG and FH have the same mid – point

Since Diagonals are equals then EFGH is a parallelogram. Hence, EFGH is a parallelogram.