Find the perimeter of a triangle whose vertices pass through ( 3, 2), (4, 5) and (6, 2) in surd form.

Analyze the question: We need the perimeter, which is the sum of the three side lengths. Each side is the distance between two vertices, so we use the distance formula.

Concept: Distance between (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Solution:

Let A = (3, 2), B = (4, 5), C = (6, 2).

• AB = √[(4 − 3)² + (5 − 2)²] = √(1 + 9) = √10
• BC = √[(6 − 4)² + (2 − 5)²] = √(4 + 9) = √13
• AC = √[(6 − 3)² + (2 − 2)²] = √9 = 3

Perimeter = 3 + √10 + √13

Tip: When two points share the same y-coordinate (or x-coordinate), the distance is just the difference of the other coordinate, so no square root is needed. Here AC = 6 − 3 = 3. Also, work out each side separately and check the numbers under the roots against the options, since the options often differ only in those values (√10 vs √12).