Question

13)\\(\triangle rst\sim\\_\\_\\_\\_\\_\\)a) similar; sss similarity; \\(\triangle egf\\)b) similar; sas similarity; \\(\triangle gef\\)c) similar; sss similarity; \\(\triangle efg\\)d) not similar

Explanation:

Step1: Calculate the ratios of corresponding sides

For \(\triangle RST\) with sides \(RS = 24\), \(ST=16\), \(RT = 12\) and \(\triangle EGF\) with sides \(EG = 6\), \(GF = 8\), \(EF=12\)
The ratio of \(RS\) to \(EG\) is \(\frac{RS}{EG}=\frac{24}{6} = 4\)
The ratio of \(ST\) to \(GF\) is \(\frac{ST}{GF}=\frac{16}{8}=2\) (This is wrong. Let's re - check the correspondence.
If we consider \(\triangle RST\) and \(\triangle EGF\) with \(RS = 24\), \(ST = 16\), \(RT=12\) and \(EG = 6\), \(GF = 8\), \(EF = 12\)
We should check the ratio of \(\frac{RS}{EF}=\frac{24}{12}=2\), \(\frac{ST}{GF}=\frac{16}{8} = 2\), \(\frac{RT}{EG}=\frac{12}{6}=2\)

Step2: Apply SSS similarity criterion

Since \(\frac{RS}{EF}=\frac{ST}{GF}=\frac{RT}{EG}=2\), by the SSS (Side - Side - Side) similarity criterion, \(\triangle RST\sim\triangle EGF\)

Answer:

A. similar; SSS similarity; \(\triangle EGF\)