Question

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in the diagram below, ( overline{ef} parallel overline{hg} ), ( ef = 5 ), ( hg )
( fi = 1.4x + 3 ), and ( hi = 6.1x - 6.5 ).

what is the length of ( overline{hi} )?

Explanation:

Step1: Use the property of similar triangles

Since \(EF\parallel HG\), \(\triangle EFI\sim\triangle HGI\). By the property of similar triangles, \(\frac{EF}{HG}=\frac{FI}{HI}\).
Substitute \(EF = 5\), \(HG=12\), \(FI = 1.4x + 3\), and \(HI=6.1x - 6.5\) into the proportion: \(\frac{5}{12}=\frac{1.4x + 3}{6.1x - 6.5}\).

Step2: Cross - multiply

Cross - multiply the proportion: \(5(6.1x-6.5)=12(1.4x + 3)\).
Expand both sides: \(30.5x-32.5 = 16.8x+36\).

Step3: Solve for \(x\)

Subtract \(16.8x\) from both sides: \(30.5x-16.8x-32.5=16.8x - 16.8x+36\), which gives \(13.7x-32.5 = 36\).
Add \(32.5\) to both sides: \(13.7x-32.5 + 32.5=36 + 32.5\), so \(13.7x=68.5\).
Divide both sides by \(13.7\): \(x=\frac{68.5}{13.7}=5\).

Step4: Find the length of \(HI\)

Substitute \(x = 5\) into the expression for \(HI\): \(HI=6.1x-6.5\).
\(HI=6.1\times5-6.5\).
\(HI = 30.5-6.5\).
\(HI=24\).

Answer:

\(24\)