Question

$\overline{ru}$ is the midsegment of the trapezoid stvw.
if $st = 5z - 41$, $ru = 4z - 16$, and $vw = -2z + 74$, what is the value of $z?$
$z = \square$

Explanation:

Step1: Apply midsegment formula

The midsegment of a trapezoid equals half the sum of the two parallel sides, so:
$$RU = \frac{ST + VW}{2}$$
Substitute the given expressions:
$$4z - 16 = \frac{(5z - 41) + (-2z + 74)}{2}$$

Step2: Simplify right-hand side

Combine like terms in the numerator:
$$4z - 16 = \frac{3z + 33}{2}$$

Step3: Eliminate denominator

Multiply both sides by 2:
$$2(4z - 16) = 3z + 33$$
$$8z - 32 = 3z + 33$$

Step4: Isolate variable z

Subtract $3z$ from both sides, then add 32 to both sides:
$$8z - 3z = 33 + 32$$
$$5z = 65$$

Step5: Solve for z

Divide both sides by 5:
$$z = \frac{65}{5}$$

Answer:

$z = 13$