Question

find the length of ef in the trapezoid shown below.
8x
7x - 1
16

Explanation:

Step1: Recall the midline theorem for trapezoids

The midline (or midsegment) of a trapezoid is parallel to the two bases and its length is the average of the lengths of the two bases. So, the formula for the midline \( EF \) (where \( EF \) is the midline, \( BC = 8x \) and \( AD = 16 \)) is \( EF=\frac{BC + AD}{2} \). We know \( EF = 7x - 1 \), so we can set up the equation:

$$ 7x-1=\frac{8x + 16}{2} $$

Step2: Solve the equation for \( x \)

First, simplify the right - hand side of the equation. \( \frac{8x + 16}{2}=4x + 8 \). So our equation becomes:

$$ 7x-1 = 4x+8 $$

Subtract \( 4x \) from both sides:

$$ 7x-4x-1=4x - 4x+8 $$

$$ 3x-1 = 8 $$

Add 1 to both sides:

$$ 3x-1 + 1=8 + 1 $$

$$ 3x=9 $$

Divide both sides by 3:

$$ x = 3 $$

Step3: Find the length of \( EF \)

We know that \( EF = 7x-1 \). Substitute \( x = 3 \) into the expression for \( EF \):

$$ EF=7\times3-1=21 - 1=20 $$

Wait, there is a mistake. Let's re - check the midline formula. The midline of a trapezoid is the average of the two bases. But in the trapezoid, \( BC = 8x \) and \( AD = 16 \), and \( EF \) is the midline. Also, from the markings, \( E \) is the mid - point of \( AB \) and \( F \) is the mid - point of \( CD \), so the midline formula is \( EF=\frac{AD + BC}{2} \). But we also can think of it as if we consider the two bases \( AD = 16 \) and \( BC = 8x \), and \( EF=7x - 1 \). So \( 7x-1=\frac{16 + 8x}{2} \).
Simplify the right - hand side: \( \frac{16 + 8x}{2}=8 + 4x \)
So the equation is \( 7x-1=4x + 8 \)
Subtract \( 4x \) from both sides: \( 3x-1 = 8 \)
Add 1 to both sides: \( 3x=9 \), so \( x = 3 \)
Then \( EF=7x-1=7\times3-1 = 20 \)? But if we check with \( BC = 8x=24 \), then the midline should be \( \frac{16 + 24}{2}=20 \), which matches. Wait, I think my first thought of a mistake was wrong. Let's confirm:
If \( AD = 16 \), \( BC = 24 \) (since \( x = 3 \), \( 8x = 24 \)), then the midline \( EF=\frac{16 + 24}{2}=20 \), and \( 7x-1=7\times3-1 = 20 \). So the length of \( EF \) is 20.

Answer:

48