Question

tw is the midsegment of the trapezoid uvxy. if xy = -z + 72, tw = 3z - 54, and uv = z + 30, what is the value of z?

Explanation:

Step1: Apply midsegment theorem

The midsegment of a trapezoid equals half the sum of the two bases, so:
$$TW = \frac{UV + XY}{2}$$

Step2: Substitute given expressions

Replace each segment with its algebraic form:
$$3z - 54 = \frac{(z + 30) + (-z + 72)}{2}$$

Step3: Simplify the right-hand side

Combine like terms in the numerator:
$$3z - 54 = \frac{102}{2}$$
$$3z - 54 = 51$$

Step4: Solve for z

Add 54 to both sides, then divide by 3:
$$3z = 51 + 54$$
$$3z = 105$$
$$z = \frac{105}{3}$$

Answer:

$z = 35$