Question

tu || sv. find uv.

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{TU} \parallel \overline{SV}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle RSV \sim \triangle RTU\). So, the ratios of corresponding sides are equal: \(\frac{RS}{RT}=\frac{RV}{RU}\).

First, find \(RS\) and \(RT\). Given \(RT = 51\), \(TS = 34\), so \(RS=RT - TS=51 - 34 = 17\)? Wait, no, wait. Wait, actually, looking at the diagram, \(RT\) is the entire side, and \(RS\) is a part? Wait, no, maybe I got it reversed. Wait, the length from \(R\) to \(T\) is \(51\)? Wait, no, the blue segment is \(RT\) with length \(51\), and \(TS\) is \(34\), so \(RS = RT - TS\)? Wait, no, maybe \(RT\) is \(51\), and \(RS\) is \(51 - 34 = 17\)? Wait, no, maybe \(RT\) is \(51\), and \(RS\) is \(34\)? Wait, no, the diagram: the red segment \(RT\) has \(TS = 34\), and the blue segment \(RT\) (the vertical one) has length \(51\). Wait, maybe the triangles are \(\triangle RSV\) and \(\triangle RTU\), so \(RS\) corresponds to \(RT\), and \(RV\) corresponds to \(RU\). Wait, let's re - examine.

Wait, the length of \(RT\) (the side from \(R\) to \(T\)): the blue segment is \(RT\) with length \(51\), and the red segment \(RT\) has \(TS = 34\), so \(RS=RT - TS\)? No, that can't be. Wait, maybe \(RT = 51\), and \(RS = 51 - 34=17\)? Wait, no, maybe the other way: \(RT\) is \(51\), and \(RS\) is \(34\), and \(ST\) is \(51 - 34 = 17\)? Wait, I think I made a mistake. Let's start over.

Wait, the two triangles: \(\triangle RSV\) and \(\triangle RTU\) are similar. So, \(\frac{RS}{RT}=\frac{RV}{RU}\). Let's denote \(RV = x\), \(RU = 69\) (since the length of \(RU\) is \(69\) as per the blue segment at the bottom). Wait, the bottom blue segment is \(RU = 69\), and \(RV\) is a part, \(UV=RU - RV\). Wait, no, we need to find \(UV\), so \(UV = RU - RV\).

First, find the ratio of \(RS\) to \(RT\). Wait, the length of \(RT\) (the vertical side) is \(51\), and \(RS\) is \(51 - 34 = 17\)? Wait, no, maybe \(RT = 51\), and \(RS = 34\), and \(ST = 51 - 34 = 17\)? No, that doesn't make sense. Wait, maybe the vertical side: \(RT = 51\), and \(RS = 34\), so the ratio \(\frac{RS}{RT}=\frac{34}{51}=\frac{2}{3}\).

Then, since \(\triangle RSV\sim\triangle RTU\), \(\frac{RV}{RU}=\frac{RS}{RT}\). We know \(RU = 69\), \(\frac{RS}{RT}=\frac{34}{51}=\frac{2}{3}\). So, \(\frac{RV}{69}=\frac{2}{3}\). Solving for \(RV\): \(RV=\frac{2}{3}\times69 = 46\). Then, \(UV=RU - RV=69 - 46 = 23\)? Wait, no, that's not right. Wait, maybe I mixed up the ratio.

Wait, let's correct the ratio. If \(\overline{SV}\parallel\overline{TU}\), then \(\triangle RSV\sim\triangle RTU\), so \(\frac{RS}{RT}=\frac{RV}{RU}\). Wait, \(RS\) is the length from \(R\) to \(S\), and \(RT\) is from \(R\) to \(T\). Given that \(RT = 51\), and \(TS = 34\), so \(RS=RT - TS = 51 - 34 = 17\). Then \(\frac{RS}{RT}=\frac{17}{51}=\frac{1}{3}\). Then, \(\frac{RV}{RU}=\frac{1}{3}\). Since \(RU = 69\), then \(RV=\frac{1}{3}\times69 = 23\). Then \(UV=RU - RV=69 - 23 = 46\)? Wait, now I'm confused. Wait, let's check the ratio again.

Wait, maybe the triangles are \(\triangle TUV\) and \(\triangle RSV\)? No, the lines are \(TU\parallel SV\), so the transversal is \(RU\) and \(RT\). So, the corresponding angles: \(\angle R\) is common, and \(\angle RSV=\angle RTU\) (corresponding angles, since \(SV\parallel TU\)), so \(\triangle RSV\sim\triangle RTU\) by AA similarity.

So, \(\frac{RS}{RT}=\frac{RV}{RU}\). Let's find \(RS\) and \(RT\). From the diagram, \(RT\) (the side from \(R\) to \(T\)) has length \(51\), and \(RS\) (the side from \(R\) to \(S\)): since \(TS = 34\), then \(RS=RT - TS=51 - 34 = 17\). So, \(\frac{RS}{RT}=\frac{17}{51}=\frac{1}{3}\). Then, \(\frac{RV}{RU}=\frac{1}{3}\). Given \(RU = 69\), then \(RV=\frac{1}{3}\times69 = 23\). Then, \(UV=RU - RV=69 - 23 = 46\)? Wait, no, that seems off. Wait, maybe I had \(RS\) and \(RT\) reversed. Maybe \(RS = 34\) and \(RT = 51\), so \(\frac{RS}{RT}=\frac{34}{51}=\frac{2}{3}\). Then, \(\frac{RV}{RU}=\frac{2}{3}\), so \(RV=\frac{2}{3}\times69 = 46\), and \(UV=RU - RV=69 - 46 = 23\). Ah, that makes more sense. Because if \(RS\) is the segment from \(R\) to \(S\) (length \(34\)) and \(RT\) is from \(R\) to \(T\) (length \(51\)), then the ratio is \(\frac{34}{51}=\frac{2}{3}\).

Step2: Set Up the Proportion

Using the similarity of \(\triangle RSV\) and \(\triangle RTU\), we have the proportion \(\frac{RS}{RT}=\frac{RV}{RU}\). We know \(RS = 34\), \(RT = 51\), \(RU = 69\). Substitute these values into the proportion: \(\frac{34}{51}=\frac{RV}{69}\).

Simplify \(\frac{34}{51}\) by dividing numerator and denominator by \(17\): \(\frac{34\div17}{51\div17}=\frac{2}{3}\). So, \(\frac{2}{3}=\frac{RV}{69}\).

Step3: Solve for \(RV\)

Cross - multiply: \(3\times RV=2\times69\), so \(3RV = 138\), then \(RV=\frac{138}{3}=46\).

Step4: Find \(UV\)

We know that \(RU = 69\) and \(RV = 46\). Since \(UV=RU - RV\), then \(UV = 69 - 46 = 23\)? Wait, no, wait. Wait, if \(RV\) is the length from \(R\) to \(V\), and \(RU\) is from \(R\) to \(U\), then \(UV=RU - RV\). But wait, maybe I got the direction wrong. Wait, maybe \(UV\) is equal to \(RV\) times a ratio? No, let's re - check.

Wait, no, the triangles: \(\triangle RSV\) and \(\triangle RTU\). So, \(RV\) corresponds to \(RU\), and \(SV\) corresponds to \(TU\), but we need \(UV\). Wait, \(RU\) is the entire base, length \(69\), \(RV\) is a part of it. So, \(UV=RU - RV\). But if \(\frac{RS}{RT}=\frac{RV}{RU}\), and \(RS = 34\), \(RT = 51\), \(RU = 69\), then \(\frac{34}{51}=\frac{RV}{69}\), \(RV=\frac{34\times69}{51}\). Simplify: \(34\) and \(51\) have a common factor of \(17\), \(34\div17 = 2\), \(51\div17 = 3\). So, \(RV=\frac{2\times69}{3}=2\times23 = 46\). Then \(UV=RU - RV=69 - 46 = 23\)? Wait, no, that can't be. Wait, maybe the triangles are \(\triangle TSV\) and \(\triangle TUV\)? No, the parallel lines are \(TU\) and \(SV\), so the correct similar triangles are \(\triangle RSV\) and \(\triangle RTU\).

Wait, another approach: Let's let \(UV = x\), then \(RV=69 - x\). Then, by Thales' theorem, \(\frac{RS}{RT}=\frac{RV}{RU}\). \(RS = 51 - 34 = 17\), \(RT = 51\), \(RV = 69 - x\), \(RU = 69\). So, \(\frac{17}{51}=\frac{69 - x}{69}\). Simplify \(\frac{17}{51}=\frac{1}{3}\), so \(\frac{1}{3}=\frac{69 - x}{69}\). Cross - multiply: \(69=3\times(69 - x)\), \(69 = 207-3x\), \(3x=207 - 69 = 138\), \(x = 46\). Ah! Here is the mistake earlier. I had \(RS\) wrong. \(RS\) is \(RT - TS=51 - 34 = 17\), not \(34\). So, \(\frac{RS}{RT}=\frac{17}{51}=\frac{1}{3}\), and \(RV = 69 - x\) (where \(x = UV\)), \(RU = 69\). So, \(\frac{17}{51}=\frac{69 - x}{69}\), \(\frac{1}{3}=\frac{69 - x}{69}\), \(69 - x=\frac{69}{3}=23\), \(x = 69 - 23 = 46\). Yes, that's correct.

So, the correct proportion is \(\frac{RS}{RT}=\frac{RV}{RU}\), where \(RS = 51 - 34 = 17\), \(RT = 51\), \(RV=69 - UV\), \(RU = 69\).

Step1 (Correct): Define Variables and Proportion

Let \(UV=x\). Then \(RV = RU - UV=69 - x\). Since \(\triangle RSV\sim\triangle RTU\) (by AA similarity, as \(\overline{SV}\parallel\overline{TU}\) implies corresponding angles are equal), we have \(\frac{RS}{RT}=\frac{RV}{RU}\).

We know \(RT = 51\), \(TS = 34\), so \(RS=RT - TS = 51 - 34=17\). \(RU = 69\), \(RV = 69 - x\).

Step2 (Correct): Substitute into Proportion

Substitute the values into the proportion: \(\frac{17}{51}=\frac{69 - x}{69}\).

Simplify \(\frac{17}{51}\) to \(\frac{1}{3}\): \(\frac{1}{3}=\frac{69 - x}{69}\).

Step3 (Correct): Solve for \(x\)

Cross - multiply: \(69=3\times(69 - x)\).

Expand the right - hand side: \(69 = 207-3x\).

Add \(3x\) to both sides: \(3x + 69 = 207\).

Subtract \(69\) from both sides: \(3x=207 - 69 = 138\).

Divide both sides by \(3\): \(x=\frac{138}{3}=46\).

Answer:

\(46\)