Explanation

To determine the nature of the points A(3, 2), B(9, 10), and C(1, 16), let's calculate the distances between each pair of points.

Step 1: Calculate the distance AB

AB = √((9 - 3)² + (10 - 2)²)

= √(6² + 8²)

= √(36 + 64)

= √100

= 10

Step 2: Calculate the distance BC

BC = √((9 - 1)² + (10 - 16)²)

= √(8² + (-6)²)

= √(64 + 36)

= √100

= 10

Step 3: Calculate the distance AC

AC = √((3 - 1)² + (2 - 16)²)

= √(2² + (-14)²)

= √(4 + 196)

= √200

= 10√2

Step 4: Determine the nature of the points

Since AB = 10, BC = 10, and AC = 10√2, and AB = BC ≠ AC, the points form an isosceles triangle.

The answer is: The points form an isosceles triangle.