Prove that the quadrilateral MNPQ is a parallelogram and find its diagonals if M (1; 1) N (6; 1) P (7; 4) Q (2; 4).

M (1; 1) N (6; 1) P (7; 4) Q (2; 4).

Find the values of the segments:

| MN | = ((6 – 1) ^ 2 + (1 – 1) ^ 2) ^ (1/2) = 5;

| NP | = ((7 – 6) ^ 2 + (4 – 1) ^ 2) ^ (1/2) = (1 + 9) ^ (1/2) = 10 ^ (1/2).

| PQ | = ((2 – 7) ^ 2 + (4 – 4) ^ 2) ^ (1/2) = 5;

| MQ | = ((2 – 1) ^ 2 + (4 – 1) ^ 2) ^ (1/2) = (1 + 9) ^ (1/2) = 10 ^ (1/2).

The opposite sides of the quadrangle are equal, which means that it is a parallelogram. Let’s find the diagonals:

| MP | = ((7 – 1) ^ 2 + (4 – 1) ^ 2) ^ (1/2) = (36 + 9) ^ (1/2) = 45 ^ (1/2).

| NQ | = ((2 – 6) ^ 2 + (4 – 1) ^ 2) ^ (1/2) = (16 + 9) ^ (1/2) = 5.