Question

find the lengths of the segments with variable expressions.
ef = 11
ad =

Explanation:

Step1: Use mid - segment theorem

In a trapezoid, the length of the mid - segment (EF) is the average of the lengths of the bases (AD and BC). The mid - segment formula is $EF=\frac{AD + BC}{2}$. Given $EF = x$, $AD=x - 4$, and $BC=2x-7$. So, $x=\frac{(x - 4)+(2x-7)}{2}$.

Step2: Cross - multiply

Multiply both sides of the equation $x=\frac{(x - 4)+(2x-7)}{2}$ by 2 to get $2x=(x - 4)+(2x-7)$.

Step3: Simplify the right - hand side

Combine like terms on the right - hand side: $2x=x - 4+2x-7$, which simplifies to $2x=3x - 11$.

Step4: Solve for x

Subtract $2x$ from both sides: $0 = 3x-2x - 11$, so $x = 11$.

Step5: Find the length of AD

Since $AD=x - 4$ and $x = 11$, then $AD=11-4$.
$AD = 7$

Answer:

7