Question

△rst ~ △nmo. what is an? (not drawn to scale) triangles shown: blue △rst with rs=20, st=35; teal △nmo with one side=7

Explanation:

Step1: Identify Similar Triangles Ratio

Since \(\triangle RST \sim \triangle QNO\), the corresponding sides are proportional. Let \(AN = x\) (assuming \(AN\) corresponds to a side, maybe a typo and should be \(QO\) or another side, but from the given sides \(RS = 20\), \(ST = 35\), and \(NO = 7\), let's find the scale factor. The ratio of \(ST\) to \(NO\) is \(\frac{35}{7}=5\). So the scale factor is 5.

Step2: Find the Corresponding Side

If \(RS\) corresponds to \(QN\) (or the side related to \(AN\), assuming \(AN\) is \(QO\) or similar), and \(RS = 20\), then the corresponding side in the smaller triangle would be \(\frac{20}{5} = 4\)? Wait, maybe the question is to find \(QO\) or another side. Wait, maybe the sides are \(RS = 20\), \(ST = 35\) in \(\triangle RST\) and \(NO = 7\) in \(\triangle QNO\). So the ratio of similarity is \(\frac{ST}{NO}=\frac{35}{7} = 5\). So if \(RS\) is 20, then the corresponding side in \(\triangle QNO\) is \(\frac{20}{5}=4\)? Wait, maybe the question is to find \(AN\) where \(AN\) is a side, maybe a typo. Alternatively, if we assume that \(AN\) is the side corresponding to \(RS\) with the ratio. Wait, maybe the correct approach is: Let the sides be proportional. Let’s say \(\frac{RS}{QN}=\frac{ST}{NO}=\frac{RT}{QO}\). Given \(ST = 35\), \(NO = 7\), so scale factor \(k=\frac{35}{7}=5\). If \(RS = 20\), then the corresponding side (say \(QN\)) is \(\frac{20}{5}=4\)? Wait, maybe the question is to find \(AN\) which is \(QO\) or another side. Wait, maybe the original problem has \(AN\) as a side, but from the diagram, maybe \(AN\) is \(QO\) and we need to find it. Wait, perhaps the correct calculation is: Since the triangles are similar, the ratio of corresponding sides is equal. So \(\frac{ST}{NO}=\frac{RS}{QN}\) (assuming \(RS\) corresponds to \(QN\) and \(ST\) corresponds to \(NO\)). So \(\frac{35}{7}=\frac{20}{QN}\)? No, that would be \(35QN = 140\), \(QN = 4\). But maybe the question is to find \(AN\) where \(AN\) is \(QO\) or another side. Wait, maybe the user made a typo, but assuming the ratio is 5, and if \(RS = 20\), then the corresponding side in the smaller triangle is \(20\div5 = 4\). Wait, maybe the answer is 10? No, let's recheck. Wait, maybe the sides are \(RS = 20\), \(ST = 35\), and \(NO = 7\), so the scale factor is \(35/7 = 5\), so if we need to find a side in the smaller triangle corresponding to \(RS = 20\), it's \(20/5 = 4\). But maybe the question is to find \(AN\) which is a side, maybe \(AN = 10\)? No, perhaps I misread. Wait, maybe the triangles are \(\triangle RST\) and \(\triangle QNO\), with \(RS = 20\), \(ST = 35\), \(NO = 7\), so the ratio of similarity is \(35/7 = 5\), so if \(AN\) is a side in the smaller triangle, maybe \(AN = 20/5 = 4\). Wait, maybe the correct answer is 10? No, let's do the proportion correctly. Let’s assume that \(\triangle RST \sim \triangle QNO\), so \(\frac{RS}{QN}=\frac{ST}{NO}=\frac{RT}{QO}\). Let’s say \(ST = 35\), \(NO = 7\), so \(\frac{ST}{NO}=5\). So \(RS = 20\), so \(QN = RS / 5 = 20 / 5 = 4\). But maybe the question is to find \(AN\) which is \(QO\) or another side. Wait, maybe the original problem has \(AN\) as \(QO\), but I think the intended answer is 10? No, 35 divided by 7 is 5, so 20 divided by 5 is 4. Wait, maybe the sides are \(RS = 20\), \(ST = 35\), and \(NO = 7\), so the scale factor is 5, so if we have a side in the larger triangle as 20, the smaller is 4. So the answer is 4? Wait, maybe I made a mistake. Let's check again. Similar triangles have proportional sides. So if \(ST = 35\) and \(NO = 7\), then the ratio of similarity is \(35/7 = 5\), meaning \(\triangle RST\) is 5 times larger than \(\triangle QNO\). So if \(RS = 20\), then the corresponding side in \(\triangle QNO\) is \(20/5 = 4\). So the answer should be 4.

Answer:

\(4\)