Question

the perimeter of a quadrilateral (four - sided polygon) is 52 inches. the longest side is four times as long as the shortest side. the other two sides are equally long and are 5 inches longer than the shortest side. find the length of all four sides.
the length of the shortest side is
the length of each of the two equal sides is
and the length of the longest side is

Explanation:

Step1: Set up the equation

Let the length of the shortest side be \(x\). Then the longest side is \(4x\), and the other two equal - length sides are \(x + 5\) each. The perimeter formula for a quadrilateral is \(P=\text{sum of all sides}\). So, \(x+(x + 5)+(x + 5)+4x=52\).

Step2: Simplify the equation

Combine like terms: \((x+x+x + 4x)+(5 + 5)=52\), which gives \(7x+10 = 52\).

Step3: Solve for \(x\)

Subtract 10 from both sides: \(7x=52 - 10\), so \(7x=42\). Then divide both sides by 7: \(x=\frac{42}{7}=6\).

Step4: Find the lengths of the other sides

The two equal - length sides: \(x + 5=6 + 5 = 11\). The longest side: \(4x=4\times6 = 24\).

Answer:

The length of the shortest side is \(6\) inches, the length of each of the two equal sides is \(11\) inches, and the length of the longest side is \(24\) inches.