Question

if \\( \triangle fgh \sim \triangle kjh \\), find \\( fh \\).

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle FGH\sim\triangle KJH\), the ratios of corresponding sides are equal. That is \(\frac{FH}{KH}=\frac{GH}{JH}\). Substituting the given values, we get \(\frac{4x - 25}{52}=\frac{x + 8}{32}\).

Step2: Cross - multiply

Cross - multiplying the equation \(\frac{4x - 25}{52}=\frac{x + 8}{32}\) gives \(32(4x-25)=52(x + 8)\).
Expanding both sides: \(128x-800 = 52x+416\).

Step3: Solve for \(x\)

Subtract \(52x\) from both sides: \(128x-52x-800=52x - 52x+416\), which simplifies to \(76x-800 = 416\).
Add \(800\) to both sides: \(76x-800 + 800=416+800\), so \(76x=1216\).
Divide both sides by \(76\): \(x=\frac{1216}{76}=16\).

Step4: Find \(FH\)

Substitute \(x = 16\) into the expression for \(FH\) (\(FH=4x-25\)). Then \(FH=4\times16-25\).
Calculate \(4\times16 = 64\), and \(64-25=39\).

Answer:

\(39\)