Question

the perimeter of the rectangle below is 172 units. find the length of side \\(\overline{bc}\\). write your answer without variables.
(there is a rectangle diagram with side \\(dc = 3x\\) and side \\(ad = 4x + 2\\), and a box for \\(bc = \square\\))

Explanation:

Step1: Write the perimeter formula for a rectangle

The perimeter formula of a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. Here, \(l=4x + 2\), \(w = 3x\), and \(P=172\). So, \(172=2((4x + 2)+3x)\).

Step2: Simplify the equation

First, simplify the expression inside the parentheses: \((4x + 2)+3x=7x + 2\). Then the equation becomes \(172 = 2(7x + 2)\). Divide both sides by 2: \(\frac{172}{2}=7x + 2\), which simplifies to \(86=7x + 2\).

Step3: Solve for \(x\)

Subtract 2 from both sides: \(86−2 = 7x\), so \(84 = 7x\). Then divide both sides by 7: \(x=\frac{84}{7}=12\).

Step4: Find the length of \(BC\)

Since \(BC = 3x\), substitute \(x = 12\) into \(3x\). So, \(BC=3\times12\).

Answer:

\(36\)