Question

if $uv = p - 21$, $wz = p$, and $xy = 2p - 31$, what is the value of $p$?
$p = \square$

Explanation:

Step1: Identify the relationship

From the diagram, we can see that \( UV + WZ = XY \) (since the segments are arranged such that the sum of \( UV \) and \( WZ \) equals \( XY \) based on the markings). So we substitute the given expressions: \( (p - 21) + p = 2p - 31 \).

Step2: Simplify the left side

Simplify \( (p - 21) + p \) to get \( 2p - 21 \). So the equation becomes \( 2p - 21 = 2p - 31 \). Wait, that can't be right. Wait, maybe I misread the diagram. Wait, actually, looking at the diagram, maybe \( XY = UV + WZ \)? Wait, no, maybe \( XY = 2 \times UV \) or something? Wait, no, the markings: \( VW = WX \) and \( UZ = ZY \), so \( WZ \) is a midline? Wait, maybe the correct relationship is that \( XY = UV + WZ \)? Wait, no, let's re-examine. Wait, \( UV = p - 21 \), \( WZ = p \), \( XY = 2p - 31 \). Wait, maybe the correct equation is \( XY = UV + WZ \)? Wait, no, that would give \( 2p - 31 = (p - 21) + p \), which simplifies to \( 2p - 31 = 2p - 21 \), which is a contradiction. So maybe the correct relationship is that \( UV = WZ \)? Wait, no, the markings: \( VW \) and \( WX \) are equal, \( UZ \) and \( ZY \) are equal. So \( WZ \) is parallel to \( UV \) and \( XY \)? Wait, maybe it's a trapezoid with midline? Wait, the midline of a trapezoid is the average of the two bases. Wait, if \( WZ \) is the midline, then \( WZ=\frac{UV + XY}{2}\). Let's try that. So \( p=\frac{(p - 21)+(2p - 31)}{2}\).

Step3: Solve the midline equation

Multiply both sides by 2: \( 2p=(p - 21)+(2p - 31) \).

Simplify the right side: \( p - 21 + 2p - 31 = 3p - 52 \).

So equation: \( 2p = 3p - 52 \).

Subtract \( 2p \) from both sides: \( 0 = p - 52 \).

Add 52 to both sides: \( p = 52 \).

Wait, let's check. If \( p = 52 \), then \( UV = 52 - 21 = 31 \), \( WZ = 52 \), \( XY = 2*52 - 31 = 104 - 31 = 73 \). Then midline \( WZ=\frac{31 + 73}{2}=\frac{104}{2}=52 \), which matches \( WZ = p = 52 \). So that works. So the correct relationship is that \( WZ \) is the midline, so \( WZ=\frac{UV + XY}{2} \).

Answer:

\( \boxed{52} \)