Question

A triangle has side lengths of (6a-3b)(6a−3b) centimeters, (5a+5c)(5a+5c) centimeters, and (8c-b)(8c−b) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?

Answer 1

P = a(61a - 36b + 50c) + 10b² + 89c² - 16bc

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Terms/Coefficients
• Expand by FOIL (First Outside Inside Last)
• Factoring

Geometry

Perimeter Formula [Triangle]: P = L₁ + L₂ + L₃

• L₁ is one side
• L₂ is another side
• L₃ is the 3rd side

Explanation:

Step 1: Define

L₁ = (6a - 3b)(6a - 3b)

L₂ = (5a + 5c)(5a + 5c)

L₃ = (8c - b)(8c - b)

Step 2: Find Perimeter

1. Substitute in variables [Perimeter - Triangle]: P = (6a - 3b)² + (5a + 5c)² + (8c - b)²
2. Expand [FOIL]: P = (36a² - 36ab + 9b²) + (25a² + 50ac + 25c²) + (b² - 16bc + 64c²)
3. Combine like terms (a²): P = 61a² - 36ab + 9b² + 50ac + 25c² + b² - 16bc + 64c²
4. Combine like terms (b²): P = 61a² + 10b² - 36ab + 50ac + 25c² - 16bc + 64c²
5. Combine like terms (c²): P = 61a² + 10b² + 89c² - 36ab + 50ac - 16bc
6. Rearrange variables: P = 61a² - 36ab + 50ac + 10b² + 89c² - 16bc
7. Factor: P = a(61a - 36b + 50c) + 10b² + 89c² - 16bc