Question

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

Explanation:

Step1: Prove similarity

Since \(\angle ECD=\angle BCA\) (vertical angles are equal) and if we assume \(\angle E=\angle B\) and \(\angle D=\angle A\) (by AA - Angle - Angle similarity criterion), \(\triangle ECD\sim\triangle BCA\).

Step2: Set up proportion

For similar triangles \(\triangle ECD\) and \(\triangle BCA\), the ratios of corresponding sides are equal. So, \(\frac{EC}{BC}=\frac{ED}{BA}\). Substituting the given values \(EC = x\), \(BC=21\), \(ED = 12\), and \(BA = 18\), we get the proportion \(\frac{x}{21}=\frac{12}{18}\).

Step3: Solve the proportion

Cross - multiply: \(18x=12\times21\). Then \(18x = 252\). Divide both sides by 18: \(x=\frac{252}{18}\). Simplify \(\frac{252\div6}{18\div6}=\frac{42}{3}=14\).

Answer:

\(x = 14\)