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 $BC=2x-7$.

Step2: Substitute into the formula

We have $x=\frac{(x - 4)+(2x-7)}{2}$. First, simplify the numerator of the right - hand side: $(x - 4)+(2x-7)=x+2x-4 - 7=3x - 11$. So the equation becomes $x=\frac{3x - 11}{2}$.

Step3: Cross - multiply

Multiply both sides of the equation by 2: $2x=3x - 11$.

Step4: Solve for x

Subtract $2x$ from both sides: $0=3x-2x - 11$, which simplifies to $x = 11$.

Step5: Find the length of AD

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

Answer:

7