Question

in the trapezoid below, ( x = ? ).

Explanation:

Step1: Apply the mid - segment formula for trapezoid

The formula for the mid - segment (or midline) of a trapezoid is \(M=\frac{a + b}{2}\), where \(M\) is the length of the mid - segment, and \(a\) and \(b\) are the lengths of the two parallel sides. Here, \(M = 5x-20\), \(a=2x + 6\), and \(b = 4x+2\). So, \(5x-20=\frac{(2x + 6)+(4x + 2)}{2}\).

Step2: Simplify the right - hand side of the equation

First, simplify \((2x + 6)+(4x + 2)\):
\((2x+6)+(4x + 2)=2x+4x+6 + 2=6x+8\).
Then the equation becomes \(5x-20=\frac{6x + 8}{2}\). Since \(\frac{6x + 8}{2}=3x + 4\), the equation is \(5x-20=3x + 4\).

Step3: Solve the linear equation for \(x\)

Subtract \(3x\) from both sides: \(5x-3x-20=3x-3x + 4\), which gives \(2x-20 = 4\).
Add \(20\) to both sides: \(2x-20+20=4+20\), so \(2x=24\).
Divide both sides by \(2\): \(x=\frac{24}{2}\).

Answer:

\(12\)