Which of the following statements is true about the points A(3, 2), B(9, 10), and C(1, 16)?
- The points form an equilateral triangle
- The points are collinear
- The points form an isosceles triangle
- None of these
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.
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