Question

ex. 3 – examine the trapezoid.
8(x + 2)
4(x − 2) 4x
7x − 8
a) find a simplified expression for the perimeter of the following, in terms of x.
b) if the perimeter is 460m, solve for x.

Explanation:

Part (a)

Step1: Recall perimeter of trapezoid

Perimeter is sum of all sides. So, add \(8(x + 2)\), \(4x\), \(7x - 8\), and \(4(x - 2)\).
\(P = 8(x + 2)+4x+(7x - 8)+4(x - 2)\)

Step2: Expand each term

Expand \(8(x + 2)=8x + 16\), \(4(x - 2)=4x - 8\).
\(P = 8x + 16+4x+7x - 8+4x - 8\)

Step3: Combine like terms

Combine \(x\)-terms: \(8x + 4x+7x + 4x = 23x\)
Combine constants: \(16 - 8 - 8 = 0\)
So, \(P = 23x\)

Step1: Set perimeter equal to 460

We know perimeter \(P = 23x\) and \(P = 460\), so \(23x = 460\)

Step2: Solve for \(x\)

Divide both sides by 23: \(x=\frac{460}{23}\)

Step3: Simplify the division

\(\frac{460}{23}=20\)

Answer:

The simplified expression for the perimeter is \(23x\).

Part (b)