Question

1. △fgh~△fbc 13. △vut~△dcb gf = ____ bc = ____

Explanation:

Step1: Use the property of similar triangles (corresponding sides are proportional) for \(\triangle FGH\sim\triangle FBC\)

Since \(\triangle FGH\sim\triangle FBC\), we have \(\frac{GF}{FC}=\frac{HF}{FB}\). Given \(HF = 21\), \(FB=15\), \(FC = 9\). Let \(GF=x\). Then \(\frac{x}{9}=\frac{21}{15}\)

Step2: Solve the proportion for \(x\)

Cross - multiply: \(15x=21\times9\), so \(x=\frac{21\times9}{15}=\frac{189}{15} = 12.6\)

Step3: Use the property of similar triangles (corresponding sides are proportional) for \(\triangle VUT\sim\triangle DCB\)

Since \(\triangle VUT\sim\triangle DCB\), we have \(\frac{UT}{CB}=\frac{VT}{DB}\). Given \(UT = 108\), \(VT = 90\), \(DB = 24\). Let \(CB = y\). Then \(\frac{108}{y}=\frac{90}{24}\)

Step4: Solve the proportion for \(y\)

Cross - multiply: \(90y=108\times24\), so \(y=\frac{108\times24}{90}=\frac{2592}{90}=28.8\)

Answer:

\(GF = 12.6\), \(BC=28.8\)