Question

6 \\( \triangle k l m - \triangle k n o \\)

Explanation:

Step1: Find the ratio of corresponding sides

Since \(\triangle KLM\sim\triangle KNO\), the ratio of their corresponding sides is equal. The ratio of \(KL\) to \(KN\) is \(\frac{KL}{KN}=\frac{10}{10 + y}\). Also, \(KM\) and \(KO\) are corresponding sides, but we can use the fact that for similar triangles \(\frac{KL}{KN}=\frac{LM}{NO}\). Given \(KL = 10\), \(LM=6\), and \(KN = 30\) (assuming \(10 + y=30\) from the figure's notation).

Step2: Solve for \(x\)

The ratio of similarity is \(\frac{KL}{KN}=\frac{10}{30}=\frac{1}{3}\). Since \(\frac{LM}{NO}=\frac{1}{3}\) (where \(LM = 6\) and \(NO=x\)), we have the proportion \(\frac{6}{x}=\frac{1}{3}\). Cross - multiply: \(x=6\times3 = 18\).

Step3: Solve for \(y\)

Since \(10 + y=30\) (from the side - length relationship in similar triangles, as \(KN\) is the longer side corresponding to \(KL\)), then \(y=30 - 10=20\).

Answer:

\(x = 18\), \(y = 20\)