Questions: A triangle has a perimeter of 53 cm. If the three sides of the triangle are n, 2n+6, and 2n+2, what is the length of each side? Separate multiple answers with a comma.

A triangle has a perimeter of 53 cm. If the three sides of the triangle are n, 2n+6, and 2n+2, what is the length of each side? Separate multiple answers with a comma.

Transcript text: A triangle has a perimeter of 53 cm . If the three sides of the triangle are $n, 2 n+6$, and $2 n+2$, what is the length of each side? Separate multiple answers with a comma.

Solution

Solution Steps

To find the length of each side of the triangle, we need to set up an equation based on the given perimeter. The sum of the three sides $ n, 2n + 6, $ and $ 2n + 2 $ should equal the perimeter, which is 53 cm. Solve this equation for $ n $ and then calculate each side using the value of $ n $.

Step 1: Set Up the Equation

To find the lengths of the sides of the triangle, we start by setting up an equation based on the given perimeter. The sides of the triangle are $ n $, $ 2n + 6 $, and $ 2n + 2 $. The perimeter is given as 53 cm. Therefore, we have the equation:

\[ n + (2n + 6) + (2n + 2) = 53 \]

Step 2: Simplify the Equation

Simplify the equation by combining like terms:

\[ n + 2n + 6 + 2n + 2 = 53 \]

This simplifies to:

\[ 5n + 8 = 53 \]

Step 3: Solve for $ n $

Solve the equation for $ n $:

\[ 5n + 8 = 53 \implies 5n = 53 - 8 \implies 5n = 45 \implies n = \frac{45}{5} = 9 \]

Step 4: Calculate the Lengths of Each Side

Using the value of $ n = 9 $, calculate the lengths of each side: