Question

solve for x. figures are not necessarily drawn to scale.
answer attempt 1 out of 2
x =
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Explanation:

Step1: Determine similarity of triangles

Since \(\angle D=\angle FGH = 82^{\circ}\) and \(\angle F\) is common to both \(\triangle FGH\) and \(\triangle FDE\), by the AA (Angle - Angle) similarity criterion, \(\triangle FGH\sim\triangle FDE\).

Step2: Set up proportion

For similar triangles \(\triangle FGH\) and \(\triangle FDE\), the ratios of corresponding sides are equal. That is \(\frac{FG}{FD}=\frac{FH}{FE}=\frac{GH}{DE}\). Let \(FD = FG + GD\), \(FE=FH + HE\). We can use \(\frac{GH}{DE}=\frac{FH}{FE}\). Substituting the given values \(GH = 4\), \(FH = 6\), \(HE=4.8\), so \(FE=6 + 4.8=10.8\). Let \(DE=x\). The proportion is \(\frac{4}{x}=\frac{6}{10.8}\).

Step3: Solve the proportion

Cross - multiply: \(6x=4\times10.8\). Then \(6x = 43.2\). Divide both sides by 6: \(x=\frac{43.2}{6}\).

Answer:

\(x = 7.2\)