Final answer:
The four points form a square as verified by using the Pythagorean theorem, which establishes that the sides are equal in length and that the diagonals are equal in length, bisect each other at right angles, and conform to the relationship for a square.
Step-by-step explanation:
To show that four points form a square, we need to prove that all sides are of equal length and that the diagonals are also of equal length and bisect each other at right angles.
According to the initial conditions:
- DA = AB = BC = CD = √32
- DB = AC = 8
Using the Pythagorean theorem which states that in a right-angled triangle:
a² + b² = c² or c = √(a² + b²)
Let's apply this to our square. In a square, diagonals bisect each other at right angles. Hence, if we consider diagonal DB, we can form two right-angled triangles by bisecting DB.
For triangle ADB, we have:
(AD)² + (AB)² = (DB)²
(√32)² + (√32)² = (8)²
32 + 32 = 64
64 = 64, which holds true.
This means that AD and AB are sides of a square and DB is a diagonal of the square. The same can be applied for the other diagonal AC, proving that all sides and diagonals are consistent with a square's properties. Therefore, the four points do indeed form a square.