Given: ABCD A (6; 7; 8) B (8; 2; 6) C (4; 3: 2) D (2; 8; 4) Determine the type of quadrilateral.

1. Given the coordinates of the vertices of the quadrilateral ABCD:

A (6; 7; 8);
B (8; 2; 6);
C (4; 3: 2);
D (2; 8; 4).
2. Find vectors AB and DC:

AB = (8; 2; 6) – (6; 7; 8) = (2; -5; -2);
DC = (4; 3: 2) – (2; 8; 4) = (2; -5; -2).
AB = DC => AB || DC.
Find vectors AD and BC:

AD = (2; 8; 4) – (6; 7; 8) = (-4; 1; -4);
BC = (4; 3: 2) – (8; 2; 6) = (-4; 1; -4).
AD = BC => AD || BC.
ABCD is a parallelogram.

3. Let’s check if ABCD is not a rectangle or a rhombus:

1) The angle DAB is not straight, so ABCD is not a rectangle:

AD * AB = (-4; 1; -4) * (2; -5; -2) = -8 – 5 + 8 = -5 ≠ 0.

2) The sides AD and AB are equal, which means that ABCD is a rhombus.

| AD | ^ 2 = (-4; 1; -4) ^ 2 = 16 + 1 + 16 = 33;
| AB | ^ 2 = (2; -5; -2) ^ 2 = 4 + 25 + 4 = 33.
| AD | = | AB |.
Answer: rhombus.