Question

if $uv = 7p + 6$, $qt = 41p - 23$, and $rs = 23p$, what is the value of $p$?
$p = $

Explanation:

Step1: Identify the relationship

From the diagram, we can see that \( UV = QT \) and \( QT + RS=UV + RS\)? Wait, no, actually, looking at the markings (the congruent segments), we can infer that \( UV + QT=RS \)? Wait, no, let's re - examine. The diagram shows that \( VQ = QU\) (the double marks) and \( QR = QS\) (the double marks)? Wait, maybe it's a trapezoid with mid - segment? Wait, no, the key is that from the segment markings, we can see that \( UV=QT\) and \( QT + UV=RS\)? Wait, no, the correct relationship here is that \( UV+QT = RS\)? Wait, no, let's look at the given expressions. Wait, actually, from the diagram, we can see that \( UV\) and \( QT\) are equal (because of the congruent marks on the sides), and \( RS=UV + QT\). Wait, \( UV = 7p + 6\), \( QT=7p + 6\) (since \( VQ = QU\) and \( QR = QS\), so \( UV=QT\)), and \( RS = 23p\), and \( QT+UV=RS\)? Wait, no, \( QT = 41p-23\), \( UV = 7p + 6\), and \( RS=23p\). Wait, maybe the correct equation is \( UV+QT=RS\)? Wait, no, let's do it properly.

Wait, the diagram has two parallel sides (the vertical sides) with \( VQ\) and \( QU\) marked congruent, and \( QR\) and \( QS\) marked congruent. So the length of \( RS\) should be equal to the sum of \( UV\) and \( QT\)? Wait, \( UV = 7p + 6\), \( QT=41p - 23\), and \( RS = 23p\). Wait, that can't be. Wait, maybe \( UV=QT\) and \( RS=UV + QT\)? Wait, no, let's set up the equation correctly.

Wait, actually, from the diagram, we can see that \( UV\) and \( QT\) are equal (because of the congruent segment markings on the non - parallel sides), and \( RS\) is equal to the sum of \( UV\) and \( QT\). Wait, \( UV=7p + 6\), \( QT = 41p-23\), and \( RS=23p\). Wait, that would give \(7p + 6+41p-23=23p\).

Step2: Solve the equation

Combine like terms on the left - hand side:
\((7p+41p)+(6 - 23)=23p\)
\(48p-17 = 23p\)

Subtract \(23p\) from both sides:
\(48p-23p-17=23p - 23p\)
\(25p-17 = 0\)

Add 17 to both sides:
\(25p-17 + 17=0 + 17\)
\(25p=17\)? No, that can't be. Wait, I made a mistake.

Wait, maybe the correct relationship is \( RS=UV + QT\) is wrong. Wait, let's look at the given values again. \( UV = 7p+6\), \( QT = 41p - 23\), \( RS=23p\). Wait, maybe \( UV=QT\) and \( RS=UV\)? No, that doesn't make sense. Wait, maybe the correct equation is \( UV+QT=RS\) is incorrect. Wait, let's re - examine the diagram. The diagram is a quadrilateral with \( U\), \( V\), \( Q\), \( R\), \( S\), \( T\). The marks on \( VQ\) and \( QU\) (double marks) and on \( QR\) and \( QS\) (double marks) suggest that \( Q\) is the mid - point of \( VR\) and \( T\) is the mid - point of \( US\). So the segment \( QT\) is a mid - segment, and the length of \( RS\) should be equal to the sum of \( UV\) and \( QT\)? Wait, no, the mid - segment theorem for trapezoids says that the length of the mid - segment is equal to the average of the lengths of the two bases. Wait, maybe \( RS=\frac{UV + QT}{1}\)? No, I think I messed up the equation.

Wait, let's start over. Let's assume that \( UV=QT\) and \( RS = UV+QT\). So \( UV = 7p + 6\), \( QT=41p-23\), so \( UV = QT\) implies \(7p + 6=41p-23\). Let's solve this equation:

Subtract \(7p\) from both sides:
\(6=41p-7p - 23\)
\(6 = 34p-23\)

Add 23 to both sides:
\(6 + 23=34p\)
\(29 = 34p\)? No, that's not right.

Wait, maybe the correct equation is \( RS=UV + QT\), so \(23p=(7p + 6)+(41p-23)\)

Step3: Solve \(23p=(7p + 6)+(41p-23)\)

First, simplify the right - hand side:
\(7p+41p+6 - 23=48p-17\)

So the equation is \(23p=48p-17\)

Subtract \(23p\) from both sides:
\(0 = 48p-23p-17\)
\(0 = 25p-17\)

Add 17 to both sides:
\(17 = 25p\)

Then \(p=\frac{17}{25}\)? No, that can't be. Wait, I must have misinterpreted the diagram.

Wait, maybe the diagram is a parallelogram? No, the markings are on the non - parallel sides. Wait, the correct relationship is that \( UV=QT\) and \( RS = UV\). No, that's not. Wait, let's check the given values again. \( UV = 7p+6\), \( QT = 41p-23\), \( RS=23p\). Maybe \( QT=RS\)? Let's set \(41p-23 = 23p\)

Subtract \(23p\) from both sides:
\(41p-23p-23=0\)
\(18p-23 = 0\)
\(18p=23\)
\(p=\frac{23}{18}\)? No.

Wait, maybe \( UV=RS\). So \(7p + 6=23p\)

Subtract \(7p\) from both sides:
\(6=23p-7p\)
\(6 = 16p\)
\(p=\frac{6}{16}=\frac{3}{8}\)? No.

Wait, I think I made a mistake in the relationship. Let's look at the diagram again. The diagram has points \(U\), \(V\), \(Q\), \(R\), \(T\), \(S\). The sides \(UV\) and \(QT\) are marked with two congruent marks (so \(UV = QT\)), and \(QR\) and \(VQ\) are marked with two congruent marks (so \(VQ=QR\)). So the length of \(RS\) should be equal to \(UV + QT\). So \(RS=UV + QT\)

\(UV = 7p + 6\), \(QT = 41p-23\), \(RS = 23p\)

So \(23p=(7p + 6)+(41p-23)\)

\(23p=48p-17\)

\(48p-23p=17\)

\(25p=17\)

\(p=\frac{17}{25}\)? No, that's not possible. Wait, maybe the equation is \(UV+RS=QT\)

\(7p + 6+23p=41p-23\)

\(30p + 6=41p-23\)

Subtract \(30p\) from both sides:

\(6 = 11p-23\)

Add 23 to both sides:

\(29 = 11p\)

\(p=\frac{29}{11}\)? No.

Wait, maybe the correct relationship is \(QT - UV=RS\)

\((41p-23)-(7p + 6)=23p\)

\(41p-23 - 7p-6=23p\)

\(34p-29 = 23p\)

Subtract \(23p\) from both sides:

\(11p-29 = 0\)

\(11p=29\)

\(p=\frac{29}{11}\)? No.

Wait, I think I made a mistake in the diagram interpretation. Let's look at the original problem again. The problem says "If \(UV = 7p + 6\), \(QT = 41p-23\), and \(RS = 23p\), what is the value of \(p\)?" and the diagram has \(V\), \(Q\), \(R\) on one side and \(U\), \(T\), \(S\) on the other. The marks on \(VQ\) and \(QU\) (double marks) and on \(QR\) and \(QS\) (double marks) mean that \(VQ = QU\) and \(QR = QS\). So the length of \(RS\) is equal to \(UV+QT\). So:

\(RS=UV + QT\)

\(23p=(7p + 6)+(41p-23)\)

\(23p=48p-17\)

\(48p-23p=17\)

\(25p=17\)

\(p=\frac{17}{25}\)? No, this can't be. Wait, maybe the equation is \(UV=RS\) and \(QT = RS\). So \(7p + 6=23p\) and \(41p-23=23p\)

From \(7p + 6=23p\), \(16p=6\), \(p=\frac{3}{8}\)

From \(41p-23=23p\), \(18p=23\), \(p=\frac{23}{18}\)

These are inconsistent. So I must have misinterpreted the diagram.

Wait, maybe the diagram is a trapezoid with \(UV\) and \(RS\) as the two bases and \(QT\) as the mid - segment. The mid - segment of a trapezoid is equal to the average of the two bases. So \(QT=\frac{UV + RS}{2}\)

So \(41p-23=\frac{(7p + 6)+23p}{2}\)

Multiply both sides by 2:

\(2(41p-23)=7p + 6+23p\)

\(82p-46 = 30p+6\)

Subtract \(30p\) from both sides:

\(52p-46 = 6\)

Add 46 to both sides:

\(52p=52\)

Divide both sides by 52:

\(p = 1\)

Ah! This makes sense. I forgot about the trapezoid mid - segment theorem. The mid - segment (or median) of a trapezoid is parallel to the two bases and its length is the average of the lengths of the two bases. So \(QT\) is the mid - segment, \(UV\) and \(RS\) are the two bases. So \(QT=\frac{UV + RS}{2}\)

Step1: Apply the trapezoid mid - segment theorem

The formula for the mid - segment \(M\) of a trapezoid with bases \(b_1\) and \(b_2\) is \(M=\frac{b_1 + b_2}{2}\). Here, \(QT\) is the mid - segment, \(UV\) and \(RS\) are the bases. So:

\(QT=\frac{UV + RS}{2}\)

Substitute \(UV = 7p + 6\), \(QT = 41p-23\), and \(RS = 23p\) into the formula:

\(41p-23=\frac{(7p + 6)+23p}{2}\)

Step2: Simplify the right - hand side

Simplify \((7p + 6)+23p=30p + 6\). So the equation becomes:

\(41p-23=\frac{30p + 6}{2}\)

Step3: Eliminate the fraction

Multiply both sides of the equation by 2 to get rid of the denominator:

\(2\times(41p-23)=30p + 6\)

\(82p-46 = 30p+6\)

Step4: Solve for \(p\)

Subtract \(30p\) from both sides:

\(82p-30p-46=30p-30p + 6\)

\(52p-46 = 6\)

Add 46 to both sides:

\(52p-46 + 46=6 + 46\)

\(52p=52\)

Divide both sides by 52:

\(p=\frac{52}{52}=1\)

Answer:

\(p = 1\)