Question

if ru = 59 and pq = 77, what is st?
st =
submit

Explanation:

Step1: Identify the figure's properties

The figure appears to have congruent segments (marked with ticks) and a right triangle or parallelogram - like structure. Notice that \( RU \) and \( ST \) might be related to \( PQ \) through a mid - segment or congruent - side relationship. Wait, actually, looking at the markings, \( U \) is the mid - point of \( TP \) (since two ticks on \( TU \) and \( UP \)) and \( R \) is the mid - point of \( QS \) (one tick on \( QR \) and \( RS \)). So, by the mid - segment theorem (or the property of trapezoids/parallelograms with mid - segments), \( ST=PQ - RU \)? Wait, no, let's think again. Wait, maybe it's a trapezoid with \( PQ \) as the longer base, \( RU \) as a mid - segment? Wait, no, the formula for the mid - segment of a trapezoid is \( \text{mid - segment}=\frac{\text{base}_1+\text{base}_2}{2} \). But here, if \( RU \) is a mid - segment and \( PQ \) and \( ST \) are the two bases, then \( RU=\frac{PQ + ST}{2} \). Wait, let's check the markings. The ticks on \( QR \) and \( RS \) mean \( QR = RS \), and ticks on \( TU \) and \( UP \) mean \( TU=UP \). So \( R \) is the mid - point of \( QS \) and \( U \) is the mid - point of \( TP \). So the segment \( RU \) connects the mid - points of \( QS \) and \( TP \), so by the mid - segment theorem for trapezoids (if \( PQ \parallel ST \) and \( QT \) and \( PS \) are the legs), then \( RU=\frac{PQ + ST}{2} \). Wait, but we know \( RU = 59 \) and \( PQ=77 \). Let's solve for \( ST \).

Step2: Solve for \( ST \) using the mid - segment formula

The mid - segment formula for a trapezoid is \( RU=\frac{PQ + ST}{2} \). We can re - arrange this formula to solve for \( ST \). Multiply both sides by 2: \( 2\times RU=PQ + ST \). Then, \( ST = 2\times RU-PQ \). Wait, no, wait: \( RU=\frac{PQ + ST}{2}\Rightarrow 2RU=PQ + ST\Rightarrow ST = 2RU - PQ \)? Wait, no, that would be if \( RU \) is between \( PQ \) and \( ST \) as a mid - segment. Wait, let's plug in the numbers. If \( RU = 59 \) and \( PQ = 77 \), then \( 2\times59=118 \), \( 118 - 77 = 41 \)? Wait, no, maybe I got the formula reversed. Wait, maybe \( PQ=\frac{RU + ST}{2} \)? No, that would not make sense. Wait, maybe the figure is a rectangle with a triangle cut off? No, the markings are for mid - points. Wait, let's start over.

Wait, the correct formula for the mid - segment (or median) of a trapezoid is \( m=\frac{a + b}{2} \), where \( m \) is the median, \( a \) and \( b \) are the two bases. So if \( RU \) is the median, and \( PQ \) and \( ST \) are the two bases, then \( RU=\frac{PQ + ST}{2} \). We know \( RU = 59 \) and \( PQ = 77 \). So:

\( 59=\frac{77+ST}{2} \)

Multiply both sides by 2: \( 59\times2=77 + ST \)

\( 118=77 + ST \)

Subtract 77 from both sides: \( ST=118 - 77=41 \)? Wait, no, that gives \( ST = 41 \)? Wait, but let's check the logic again. Wait, maybe \( PQ \) is the top base and \( ST \) is the bottom base, and \( RU \) is the middle. Wait, maybe I had the formula reversed. If \( PQ \) is longer than \( RU \), then maybe \( PQ=RU + ST \)? No, that doesn't fit the mid - segment formula. Wait, the mid - segment is the average of the two bases. So if \( RU \) is the mid - segment, then \( RU=\frac{PQ + ST}{2} \). So solving for \( ST \):

\( ST = 2RU - PQ \)

Plugging in \( RU = 59 \) and \( PQ = 77 \):

\( ST=2\times59 - 77=118 - 77 = 41 \)? Wait, no, 2*59 is 118, 118 - 77 is 41? Wait, but let's check the figure again. The figure has \( Q \), \( R \), \( S \) on one side, and \( P \), \( U \), \( T \) on the other. The ticks on \( QR \) and \( RS \) mean \( QR = RS \), and ticks on \( TU \) and \( UP \) mean \( TU = UP \). So \( R \) is the mid - point of \( QS \) and \( U \) is the mid - point of \( TP \). So the segment \( RU \) connects mid - points, so by the mid - segment theorem for trapezoids (if \( PQ \parallel ST \)), \( RU=\frac{PQ + ST}{2} \). So we can solve for \( ST \):

\( 59=\frac{77 + ST}{2} \)

Multiply both sides by 2: \( 118=77 + ST \)

Subtract 77: \( ST=118 - 77 = 41 \)

Answer:

\( 41 \)