A straight line passes through the point P(1,2) and Q (5,8). Calculate the length PQ

Analyze the question: We need the distance between two points, P(1, 2) and Q(5, 8).

Concept: The distance formula comes from Pythagoras' theorem:

PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]

Solution:

• Horizontal difference: 5 − 1 = 4
• Vertical difference: 8 − 2 = 6

PQ = √(4² + 6²) PQ = √(16 + 36) PQ = √52 PQ = √(4 × 13) PQ = 2√13

Tip: After finding the square root, always check for a perfect-square factor (4, 9, 16, 25...) to simplify. Here 52 = 4 × 13, so √52 = 2√13. Also, a 4-6 right triangle is just a 2-3 triangle scaled by 2, so the hypotenuse is 2 × √(2² + 3²) = 2√13. This is a quick shortcut.