Question

a triangle has side lengths of (4m + 3n) centimeters, (8m + 10p) centimeters, and (2p + 2n) centimeters. which expression represents the perimeter, in centimeters, of the triangle? answer options: 5n + 12p + 12m; 4np + 18mp + 7mn; 15np + 14mp; 4p + 12m + 13n

Explanation:

Step1: Recall perimeter formula

Perimeter of a triangle is the sum of its three sides. So, we need to add \((4m + 3n)\), \((8m + 10p)\), and \((2p + 2n)\).
\[
(4m + 3n)+(8m + 10p)+(2p + 2n)
\]

Step2: Combine like terms for \(m\)

The terms with \(m\) are \(4m\) and \(8m\). Adding them: \(4m + 8m=12m\)

Step3: Combine like terms for \(n\)

The terms with \(n\) are \(3n\) and \(2n\). Adding them: \(3n + 2n = 5n\)

Step4: Combine like terms for \(p\)

The terms with \(p\) are \(10p\) and \(2p\). Adding them: \(10p+2p = 12p\)

Step5: Combine all results

Putting the combined terms together, we get \(12m + 5n+12p\) (or \(5n + 12p+12m\) as addition is commutative).

Answer:

\(5n + 12p + 12m\) (the first option)