Question

solve for x. figures are not necessarily drawn to scale.

Explanation:

Step1: Use the property of similar triangles

Since \(\angle E=\angle GHF = 90^{\circ}\) and \(\angle F\) is common to \(\triangle DEF\) and \(\triangle HGF\), \(\triangle DEF\sim\triangle HGF\) by the AA (Angle - Angle) similarity criterion.
For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{DE}{GH}=\frac{EF}{HF}\)

Step2: Substitute the given values

We know that \(DE = 42\), \(EF=28 + 21=49\), \(HF = 21\), and \(GH=x\)
Substituting into \(\frac{DE}{GH}=\frac{EF}{HF}\), we get \(\frac{42}{x}=\frac{49}{21}\)

Step3: Cross - multiply

Cross - multiplying gives \(49x=42\times21\)

Step4: Solve for \(x\)

First, calculate \(42\times21 = 882\). Then \(x=\frac{882}{49}=18\)

Answer:

\(18\)