Given vectors A (4; -2), B (2; 4), C (0; -2) Tasks: 1) Find the lengths of the sides

Given vectors A (4; -2), B (2; 4), C (0; -2) Tasks: 1) Find the lengths of the sides 2) The angles of the triangle 3) The length of the median AM 4) The length of the median line parallel to AB.

1) Find the sides of the triangle:

| AB | = ((2 – 4) ^ 2 + (4 + 2) ^ 2) ^ (1/2) = 40 ^ (1/2);

| AC | = ((0 – 4) ^ 2 + (-2 + 2) ^ 2) ^ (1/2) = 4;

| BC | = ((0 – 2) ^ 2 + (-2 – 4) ^ 2) ^ (1/2) = 40 ^ (1/2);

2) We use the cosine theorem for the side AC:

16 = 40 + 40 – 2 * 40 * cos B;

16 = 80 – 80 * cos B;

80 * cos B = 64;

cos B = 4/5;

B = 36 °;

A = C = 72 °.

3) M is the midpoint of the segment BC, so it has coordinates:

M (1; 1).

| AM | = ((4 – 1) ^ 2 + (-2 – 1) ^ 2) ^ (1/2) = 18 ^ (1/2).

4) The middle line parallel to AB is equal to half of it, that is, 10 ^ (1/2).