Correct Answer:
A. ”√32″
The distance between two points in a coordinate plane is found using the distance formula, derived from the Pythagorean theorem.
- Correct Answer: √32. The distance formula is d = √[(x₂-x₁)² + (y₂-y₁)²]. Given points A(p,0) and B(0,p), and distance d=8: 8 = √[(0-p)² + (p-0)²]. This simplifies to 8 = √[(-p)² + (p)²] = √[p² + p²] = √[2p²]. Squaring both sides gives 64 = 2p², so p² = 32. Therefore, p = √32.
- Distractor B: 4√6 is incorrect; √32 simplifies to 4√2.
- Distractor C: 4 is incorrect; if p=4, the distance is √32, not 8.
- Distractor D: √4 (which is 2) is incorrect; if p=2, the distance is √8, not 8.
Accurate application of the distance formula is crucial for solving coordinate geometry problems.