Question

q and t are the midpoints of the legs, \\(\overline{rv}\\) and \\(\overline{su}\\), of trapezoid rsvu. if \\(uv = y - 12\\), \\(qt = -y + 66\\), and \\(rs = y\\), what is the value of \\(y\\)?

Explanation:

Step1: Recall trapezoid midsegment formula

The midsegment (median) of a trapezoid is the average of the lengths of the two bases, so $QT = \frac{RS + UV}{2}$

Step2: Substitute given values

Substitute $UV = y - 12$, $QT = -y + 66$, $RS = y$ into the formula:
$$-y + 66 = \frac{y + (y - 12)}{2}$$

Step3: Simplify the right-hand side

Combine like terms in the numerator:
$$-y + 66 = \frac{2y - 12}{2}$$
$$-y + 66 = y - 6$$

Step4: Solve for $y$

Rearrange terms to isolate $y$:
$$66 + 6 = y + y$$
$$72 = 2y$$
$$y = \frac{72}{2} = 48$$

Answer:

$y=48$