Question

1. $\triangle agn \sim \triangle fls$; find $x$

Explanation:

Step1: Set up proportion (similar triangles)

Since \(\triangle AGN \sim \triangle FLS\), the corresponding sides are proportional. So \(\frac{AG}{FL}=\frac{GN}{LS}\). Substituting the given lengths: \(\frac{24}{x}=\frac{x + 7}{6}\)

Step2: Cross - multiply

Cross - multiplying gives \(24\times6=x(x + 7)\)
\(144=x^{2}+7x\)

Step3: Rearrange into quadratic equation

Rearranging the equation to standard quadratic form: \(x^{2}+7x - 144 = 0\)

Step4: Factor the quadratic equation

Factor the quadratic: \(x^{2}+16x - 9x - 144 = 0\)
\(x(x + 16)-9(x + 16)=0\)
\((x + 16)(x - 9)=0\)

Step5: Solve for x

Setting each factor equal to zero: \(x+16 = 0\) or \(x - 9 = 0\). Since length cannot be negative, we discard \(x=-16\). So \(x = 9\)

Answer:

\(x = 9\)