Question

ux is the midsegment of the trapezoid stvw. if vw = 48z - 80, ux = 41z - 60, and st = 16z - 4, what is the value of z? z =

Explanation:

Step1: Recall midsegment formula for trapezoid

The midsegment of a trapezoid is the average of the two bases. So, \( UX=\frac{ST + VW}{2} \).
Substitute the given expressions: \( 41z - 60=\frac{(16z - 4)+(48z - 80)}{2} \).

Step2: Simplify the right - hand side

First, combine like terms in the numerator: \( (16z - 4)+(48z - 80)=16z+48z-4 - 80 = 64z-84 \).
Then, divide by 2: \( \frac{64z - 84}{2}=32z-42 \).
Now our equation is \( 41z - 60 = 32z-42 \).

Step3: Solve for z

Subtract \( 32z \) from both sides: \( 41z-32z - 60=32z-32z-42 \), which simplifies to \( 9z - 60=-42 \).
Add 60 to both sides: \( 9z-60 + 60=-42 + 60 \), so \( 9z=18 \).
Divide both sides by 9: \( z=\frac{18}{9}=2 \).

Answer:

\( z = 2 \)