Question

1. the perimeter of the polygon is 18 yards. find the missing side.
2. the perimeter of the polygon is ( 3x^{3}+2x + 2 ). find a polynomial that represents the length of the missing side.

Explanation:

Step1: Recall the formula for perimeter

The perimeter \(P\) of a polygon is the sum of all its side - lengths. Let the missing side be \(s\). For the first polygon, \(P = 2 + 5+9 + s\). Given \(P = 18\).

Step2: Solve for the missing side of the first polygon

Substitute the values into the perimeter formula:
\(18=2 + 5+9 + s\)
First, calculate \(2 + 5+9=16\). Then the equation becomes \(18=16 + s\).
Subtract 16 from both sides: \(s=18 - 16\)
\(s = 2\) yards.

For the second polygon, assume the other two non - missing sides are also \(x\) (since the figure is a quadrilateral and two sides are given as \(x\)). Let the missing side be \(m\). The perimeter formula is \(P=x + x+m\). Given \(P = 3x^{3}+2x + 2\).

Step3: Solve for the missing side of the second polygon

Substitute into the formula: \(3x^{3}+2x + 2=x + x+m\)
Simplify the right - hand side: \(3x^{3}+2x + 2=2x+m\)
Subtract \(2x\) from both sides: \(m=3x^{3}+2x + 2-2x\)
\(m = 3x^{3}+2\)

Answer:

1. The missing side is \(2\) yards.
2. The polynomial representing the missing side is \(3x^{3}+2\)