Question

5 △fgh-△igj

Explanation:

Step1: Use the property of similar triangles

For similar triangles \(\triangle FGH\) and \(\triangle IGJ\), the ratios of corresponding sides are equal.

Step2: Set up the proportion for side \(x\)

We have \(\frac{FG}{FI + FG}=\frac{GH}{GH + HJ}\), substituting the values \(FG = 6\), \(FI=2\), \(GH = 5\), we get \(\frac{6}{6 + 2}=\frac{5}{5 + x}\).
Cross - multiply: \(6(5 + x)=8\times5\).
Expand: \(30+6x = 40\).
Subtract 30 from both sides: \(6x=40 - 30=10\).
Solve for \(x\): \(x=\frac{10}{6}=\frac{5}{3}\).

Step3: Set up the proportion for side \(y\)

Since \(\frac{GH}{HJ}=\frac{FG}{FI}\), and \(y = GH+HJ\), \(HJ=x=\frac{5}{3}\), \(GH = 5\).
First, \(\frac{5}{\frac{5}{3}}=\frac{6}{2}=3\) (checking the ratio). Then \(y=5+\frac{5}{3}=\frac{15 + 5}{3}=\frac{20}{3}\).

Answer:

\(x = \frac{5}{3}\), \(y=\frac{20}{3}\)