Question

if $st = p - 1$, $ux = -6p + 58$, and $vw = -5p + 53$, what is the value of $p$?
$p = \square$

Explanation:

Step1: Identify the midsegment theorem

From the diagram, \(UX\) and \(ST\) are related to \(VW\) by the midsegment theorem (or trapezoid midsegment property), where \(UX=\frac{ST + VW}{2}\).

Step2: Substitute the given expressions

Substitute \(ST = p - 1\), \(UX=-6p + 58\), and \(VW=-5p + 53\) into the formula:
\(-6p + 58=\frac{(p - 1)+(-5p + 53)}{2}\)

Step3: Simplify the right - hand side

First, simplify the numerator of the right - hand side: \((p - 1)+(-5p + 53)=p-1-5p + 53=-4p + 52\)
Then the equation becomes \(-6p + 58=\frac{-4p + 52}{2}\)
Simplify the right - hand side: \(\frac{-4p + 52}{2}=-2p+26\)
So the equation is \(-6p + 58=-2p + 26\)

Step4: Solve for \(p\)

Add \(6p\) to both sides of the equation:
\(-6p+6p + 58=-2p+6p + 26\)
\(58 = 4p+26\)
Subtract 26 from both sides:
\(58 - 26=4p+26 - 26\)
\(32 = 4p\)
Divide both sides by 4:
\(p=\frac{32}{4}=8\)

Answer:

\(8\)