Question

in the diagram, \\( \triangle def \sim \triangle ghi \\).

what is the value of \\( x \\)?

\\( \bigcirc 2.9 \\)
\\( \bigcirc 3.5 \\)
\\( \bigcirc 4.25 \\)
\\( \bigcirc 7.17 \\)

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle DEF\sim\triangle GHI\), the ratios of corresponding sides are equal. The side \(DF = 6\) in \(\triangle DEF\) corresponds to the side \(GI=(2x - 1)\) in \(\triangle GHI\), and the side \(DE = 10\) in \(\triangle DEF\) corresponds to the side \(GH = 8\) in \(\triangle GHI\). So, we have the proportion \(\frac{DF}{GI}=\frac{DE}{GH}\).

Step2: Substitute the values into the proportion

Substitute \(DF = 6\), \(GI=(2x - 1)\), \(DE = 10\), and \(GH = 8\) into \(\frac{DF}{GI}=\frac{DE}{GH}\), we get \(\frac{6}{2x-1}=\frac{10}{8}\).

Step3: Cross - multiply

Cross - multiplying gives \(10\times(2x - 1)=6\times8\).
Expand the left side: \(20x-10 = 48\).

Step4: Solve for \(x\)

Add \(10\) to both sides: \(20x=48 + 10\), so \(20x=58\).
Divide both sides by \(20\): \(x=\frac{58}{20}=2.9\).

Answer:

\(2.9\)