Question

△pqr ~ △pfg
in the figure above, the two lines appearing parallel are parallel.
find the length of \\(\overline{fq}\\):
answer

Explanation:

Step1: Use Similar Triangles Proportion

Since \(\triangle PQR \sim \triangle PFG\), the corresponding sides are proportional. Let \(FQ = x\). Then \(PQ = x + 54\). The sides \(PR = 91\), \(PG = 78\), \(PF = 54\), and \(PQ = x + 54\), \(PF = 54\), \(PR = 91\), \(PG = 78\). So the proportion is \(\frac{PQ}{PF}=\frac{PR}{PG}\), which is \(\frac{x + 54}{54}=\frac{91}{78}\).

Step2: Solve the Proportion

Cross - multiply: \(78(x + 54)=54\times91\)
First, simplify the right - hand side: \(54\times91 = 4914\)
Left - hand side: \(78x+78\times54=78x + 4212\)
So the equation is \(78x+4212 = 4914\)
Subtract 4212 from both sides: \(78x=4914 - 4212=702\)
Divide both sides by 78: \(x=\frac{702}{78}=9\) Wait, no, wait. Wait, maybe I mixed up the sides. Wait, actually, the correct proportion should be \(\frac{PQ}{PF}=\frac{PR}{PG}\), but let's re - check. Wait, \(\triangle PQR\) and \(\triangle PFG\) are similar, so \(\frac{PQ}{PF}=\frac{PR}{PG}\). Wait, \(PQ\) corresponds to \(PF\)? No, wait, the correspondence: \(\triangle PQR\sim\triangle PFG\), so vertex \(P\) corresponds to \(P\), \(Q\) corresponds to \(F\), \(R\) corresponds to \(G\). So the sides: \(PQ\) corresponds to \(PF\), \(QR\) corresponds to \(FG\), \(PR\) corresponds to \(PG\). Wait, no, that's wrong. Wait, if the triangles are similar, the order of the letters matters. So \(\triangle PQR\sim\triangle PFG\) means \(P ightarrow P\), \(Q ightarrow F\), \(R ightarrow G\). So \(PQ\) is corresponding to \(PF\), \(QR\) to \(FG\), \(PR\) to \(PG\). Wait, but the parallel lines are \(QR\) and \(FG\), so \(\angle Q=\angle F\) (corresponding angles), \(\angle R=\angle G\) (corresponding angles), so the similarity is correct as \(\triangle PQR\sim\triangle PFG\) with \(P\) as the common vertex. So the sides: \(PQ\) (from \(P\) to \(Q\)) and \(PF\) (from \(P\) to \(F\)) are corresponding, \(PR\) (from \(P\) to \(R\)) and \(PG\) (from \(P\) to \(G\)) are corresponding. So the proportion is \(\frac{PQ}{PF}=\frac{PR}{PG}\). Let \(FQ = x\), then \(PQ=PF + FQ=54 + x\), \(PF = 54\), \(PR = 91\), \(PG = 78\). So \(\frac{54 + x}{54}=\frac{91}{78}\)
Simplify \(\frac{91}{78}=\frac{13}{12}\) (dividing numerator and denominator by 7). So \(\frac{54 + x}{54}=\frac{13}{12}\)
Cross - multiply: \(12(54 + x)=54\times13\)
\(12\times54+12x = 702\)
\(648+12x = 702\)
\(12x=702 - 648 = 54\)
\(x=\frac{54}{12}=4.5\)? No, that's not right. Wait, I think I mixed up the correspondence. Let's do it again. The correct correspondence: Since \(QR\parallel FG\), then \(\triangle PQR\sim\triangle PFG\) with \(P\) as the common vertex, so the sides \(PR\) and \(PG\) are corresponding, \(PQ\) and \(PF\) are corresponding, and \(QR\) and \(FG\) are corresponding. Wait, but \(PR = 91\), \(PG = 78\), \(PF = 54\), and \(PQ=PF + FQ=54 + FQ\). So the proportion should be \(\frac{PR}{PG}=\frac{PQ}{PF}\), so \(\frac{91}{78}=\frac{54 + FQ}{54}\)
Simplify \(\frac{91}{78}=\frac{13}{12}\)
So \(\frac{13}{12}=\frac{54 + FQ}{54}\)
Multiply both sides by 54: \(54\times\frac{13}{12}=54 + FQ\)
\(54\times\frac{13}{12}=\frac{54\times13}{12}=\frac{702}{12}=58.5\)
Then \(58.5=54 + FQ\)
Subtract 54: \(FQ = 58.5 - 54 = 4.5\)? No, that can't be. Wait, maybe the correspondence is \(\frac{PF}{PQ}=\frac{PG}{PR}\). Let's try that. So \(\frac{54}{PQ}=\frac{78}{91}\)
Simplify \(\frac{78}{91}=\frac{6}{7}\) (dividing numerator and denominator by 13)
So \(\frac{54}{PQ}=\frac{6}{7}\)
Cross - multiply: \(6PQ = 54\times7\)
\(6PQ = 378\)
\(PQ=\frac{378}{6}=63\)
Then \(FQ=PQ - PF=63 - 54 = 9\). Ah, there we go. I had the proportion reversed. The correct proportion is \(\frac{PF}{PQ}=\frac{PG}{PR}\) because in similar triangles, the ratio of corresponding sides. Since \(\triangle PFG\sim\triangle PQR\) (order matters), so \(PF\) corresponds to \(PQ\), \(PG\) corresponds to \(PR\), \(FG\) corresponds to \(QR\). So \(\frac{PF}{PQ}=\frac{PG}{PR}\)

So \(\frac{54}{PQ}=\frac{78}{91}\)
Simplify \(\frac{78}{91}=\frac{6}{7}\) (dividing numerator and denominator by 13)
So \(\frac{54}{PQ}=\frac{6}{7}\)
Cross - multiply: \(6PQ=54\times7\)
\(6PQ = 378\)
\(PQ = 63\)
Since \(PQ=PF + FQ\) and \(PF = 54\), then \(FQ=PQ - PF=63 - 54 = 9\)? Wait, no, 63 - 54 is 9? Wait, 54+9 = 63. But let's check the proportion again. If \(PQ = 63\), \(PF = 54\), \(PR = 91\), \(PG = 78\), then \(\frac{PF}{PQ}=\frac{54}{63}=\frac{6}{7}\), and \(\frac{PG}{PR}=\frac{78}{91}=\frac{6}{7}\). Yes, that's correct. So the length of \(FQ\) is 9? Wait, no, wait, 63 - 54 is 9? Wait, maybe I made a mistake in the correspondence. Wait, let's start over.

Given \(\triangle PQR\sim\triangle PFG\), so the ratio of corresponding sides is equal. Let's identify the corresponding sides. Since \(QR\parallel FG\), \(\angle QPR=\angle FPG\) (common angle), \(\angle PQR=\angle PFG\) (corresponding angles), \(\angle PRQ=\angle PGF\) (corresponding angles). So the correspondence is \(P ightarrow P\), \(Q ightarrow F\), \(R ightarrow G\). Therefore, the sides: \(PQ\) corresponds to \(PF\), \(QR\) corresponds to \(FG\), \(PR\) corresponds to \(PG\). So the ratio is \(\frac{PQ}{PF}=\frac{PR}{PG}\). Let \(FQ = x\), then \(PQ=PF + FQ=54 + x\), \(PF = 54\), \(PR = 91\), \(PG = 78\). So \(\frac{54 + x}{54}=\frac{91}{78}\)
Simplify \(\frac{91}{78}=\frac{13}{12}\)
So \(\frac{54 + x}{54}=\frac{13}{12}\)
Multiply both sides by 54: \(54 + x=\frac{13}{12}\times54\)
\(\frac{13}{12}\times54=\frac{13\times54}{12}=\frac{13\times9}{2}=\frac{117}{2}=58.5\)
Then \(x=58.5 - 54 = 4.5\). Wait, now I'm confused. Which is correct?

Wait, let's calculate \(\frac{91}{78}\): \(91\div13 = 7\), \(78\div13 = 6\), so \(\frac{91}{78}=\frac{7}{6}\). Oh! I made a mistake in simplifying \(\frac{91}{78}\). \(91 = 13\times7\), \(78 = 13\times6\), so \(\frac{91}{78}=\frac{7}{6}\), not \(\frac{13}{12}\). That's the error!

So let's redo the proportion with the correct simplification.

\(\frac{PQ}{PF}=\frac{PR}{PG}\), \(PQ = 54 + x\), \(PF = 54\), \(PR = 91\), \(PG = 78\)

\(\frac{54 + x}{54}=\frac{91}{78}=\frac{7}{6}\)

Cross - multiply: \(6(54 + x)=54\times7\)

\(324+6x = 378\)

Subtract 324 from both sides: \(6x=378 - 324 = 54\)

Divide by 6: \(x = 9\)

Ah, because \(\frac{91}{78}=\frac{7}{6}\) (since \(91\div13 = 7\), \(78\div13 = 6\)). So the correct value of \(x\) (which is \(FQ\)) is 9? Wait, no, \(54 + x\) is \(PQ\), and when we solve \(6(54 + x)=54\times7\), \(54\times7 = 378\), \(6\times54=324\), \(378 - 324 = 54\), \(x = 9\). So \(FQ = 9\)? Wait, but let's check with the other ratio. \(\frac{PR}{PG}=\frac{91}{78}=\frac{7}{6}\), \(\frac{PQ}{PF}=\frac{54 + 9}{54}=\frac{63}{54}=\frac{7}{6}\). Yes, that's correct. So the length of \(FQ\) is 9? Wait, no, 63 - 54 is 9, so \(FQ = 9\).

Answer:

\(9\)