Question

show that the triangles are similar and write a similarity statement.
so \\( \triangle g h j \sim \triangle \square \\)

Explanation:

Step1: Calculate the ratios of corresponding sides

• Calculate \(\frac{HG}{FG}\): \(HG = 15\), \(FG=5\), so \(\frac{HG}{FG}=\frac{15}{5} = 3\)
• Calculate \(\frac{HJ}{FK}\): \(HJ = 16.5\), \(FK=16.5 + 5.5=22\), so \(\frac{HJ}{FK}=\frac{16.5}{22}=\frac{16.5\div5.5}{22\div5.5}=\frac{3}{4}\) (Wait, no. Wait, actually \(FK = 16.5+5.5 = 22\) is wrong. Wait, for the ratio of sides. The sides for similarity. We should use \(\frac{HG}{HF}\), \(HF=15 + 5=20\), \(\frac{HG}{HF}=\frac{15}{20}=\frac{3}{4}\); \(\frac{HJ}{HK}\), \(HK=16.5+5.5 = 22\), \(\frac{HJ}{HK}=\frac{16.5}{22}=\frac{3}{4}\); \(\frac{GJ}{FK}\), \(FK = 24\), \(\frac{GJ}{FK}=\frac{18}{24}=\frac{3}{4}\)

Step2: Use the Side - Side - Side (SSS) similarity criterion

Since \(\frac{HG}{HF}=\frac{HJ}{HK}=\frac{GJ}{FK}=\frac{3}{4}\), by the SSS (Side - Side - Side) similarity criterion, the two triangles are similar.

Answer:

\(\frac{HG}{HF}=\frac{HJ}{HK}=\frac{GJ}{FK}\), so \(\triangle GHJ\sim\triangle FHK\)