Question

if $pq = 3v - 63$, $ru = v$, and $st = 4v - 72$, what is the value of $v$?
$v = \square$

Explanation:

Step1: Identify Midsegment Theorem

In a trapezoid, the midsegment (or median) is parallel to the bases and its length is the average of the lengths of the two bases. Here, \( RU \) is the midsegment, \( PQ \) and \( ST \) are the bases. So, \( RU=\frac{PQ + ST}{2} \).

Step2: Substitute Given Expressions

Substitute \( PQ = 3v - 63 \), \( RU = v \), and \( ST = 4v - 72 \) into the formula:
\( v=\frac{(3v - 63)+(4v - 72)}{2} \)

Step3: Simplify the Equation

Multiply both sides by 2:
\( 2v=(3v - 63)+(4v - 72) \)
Combine like terms on the right:
\( 2v = 7v-135 \)
Subtract \( 7v \) from both sides:
\( - 5v=-135 \)

Step4: Solve for \( v \)

Divide both sides by \( - 5 \):
\( v = 27 \)

Answer:

\( 27 \)