Question

1. △smr is similar to △sat. find the length of \\(\overline{st}\\).

Explanation:

Step1: Set up proportion for similar triangles

Since \(\triangle SMR \sim \triangle SAT\), the ratios of corresponding sides are equal. So \(\frac{SM}{SA}=\frac{SR}{ST}\). Wait, actually, let's identify corresponding sides. \(SM\) corresponds to \(SA\), \(MR\) corresponds to \(AT\), and \(SR\) corresponds to \(ST\)? Wait, no, let's check the sides. \(MR = 3.6\), \(AT = 4.5\), \(SR = 4\), and we need \(ST\). So the ratio of \(MR\) to \(AT\) should equal the ratio of \(SR\) to \(ST\)? Wait, no, similar triangles: corresponding sides. Let's see, \(\triangle SMR\) and \(\triangle SAT\), so angle at \(S\) is common, so sides around angle \(S\): \(SM\) and \(SA\), \(SR\) and \(ST\), and \(MR\) and \(AT\). So \(\frac{MR}{AT}=\frac{SR}{ST}\)? Wait, no, \(\frac{SM}{SA}=\frac{MR}{AT}=\frac{SR}{ST}\). Wait, \(MR = 3.6\), \(AT = 4.5\), \(SR = 4\), let's let \(ST = x\). Then \(\frac{3.6}{4.5}=\frac{4}{x}\)? Wait, no, maybe \(\frac{SR}{ST}=\frac{MR}{AT}\). Wait, let's confirm: \(\triangle SMR \sim \triangle SAT\), so \(SM\) corresponds to \(SA\), \(MR\) corresponds to \(AT\), \(SR\) corresponds to \(ST\). So the ratio of \(MR\) to \(AT\) is equal to the ratio of \(SR\) to \(ST\). So \(\frac{MR}{AT}=\frac{SR}{ST}\). Plugging in values: \(MR = 3.6\), \(AT = 4.5\), \(SR = 4\), \(ST = x\). So \(\frac{3.6}{4.5}=\frac{4}{x}\)? Wait, no, that would be if \(SR\) and \(ST\) are corresponding, but maybe it's \(\frac{SR}{ST}=\frac{MR}{AT}\)? Wait, let's solve for \(x\). Cross - multiply: \(3.6x = 4.5\times4\). Wait, no, wait, maybe I mixed up. Wait, actually, \(\frac{SM}{SA}=\frac{MR}{AT}=\frac{SR}{ST}\). But \(SM\) and \(SA\): \(SM\) is part of \(SA\)? Wait, no, looking at the diagram, \(M\) is on \(SA\), and \(R\) is on \(ST\). So \(SA = SM + MA\)? Wait, no, the diagram shows \(M\) on \(SA\) and \(R\) on \(ST\), so \(SM\) is a segment from \(S\) to \(M\) on \(SA\), \(MR\) is parallel to \(AT\) (since triangles are similar, so \(MR \parallel AT\)). So the ratio of \(SM\) to \(SA\) is equal to the ratio of \(MR\) to \(AT\) and equal to the ratio of \(SR\) to \(ST\). So \(\frac{MR}{AT}=\frac{SR}{ST}\). So \(MR = 3.6\), \(AT = 4.5\), \(SR = 4\), \(ST = x\). So \(\frac{3.6}{4.5}=\frac{4}{x}\)? Wait, no, that would be \(\frac{3.6}{4.5}=\frac{4}{x}\), but let's check the ratio \(\frac{3.6}{4.5}\) simplifies to \(\frac{36}{45}=\frac{4}{5}\). Wait, \(\frac{3.6}{4.5}=\frac{36}{45}=\frac{4}{5}\). So \(\frac{4}{5}=\frac{4}{x}\)? That would give \(x = 5\)? Wait, no, wait, maybe I had the ratio reversed. Let's do it correctly: \(\triangle SMR \sim \triangle SAT\), so \(\frac{SA}{SM}=\frac{AT}{MR}=\frac{ST}{SR}\). Ah, there we go! I had the ratio reversed. So \(SA\) corresponds to \(SM\), \(AT\) corresponds to \(MR\), \(ST\) corresponds to \(SR\). So \(\frac{AT}{MR}=\frac{ST}{SR}\). That makes more sense. So \(AT = 4.5\), \(MR = 3.6\), \(ST = x\), \(SR = 4\). So \(\frac{4.5}{3.6}=\frac{x}{4}\).

Step2: Solve for \(x\) (length of \(ST\))

Cross - multiply: \(3.6x = 4.5\times4\). Calculate \(4.5\times4 = 18\). Then \(x=\frac{18}{3.6}\). Divide \(18\) by \(3.6\): \(18\div3.6 = 5\). Wait, let's check: \(\frac{4.5}{3.6}=\frac{5}{4}\)? Wait, \(4.5\div3.6 = 1.25\), and \(5\div4 = 1.25\). Yes, that works. So \(ST = 5\).

Answer:

\(5\)