Question

1. if \\( \triangle h b n \sim \triangle l y r \\), find the value of \\( x \\).

Explanation:

Step1: Use the property of similar triangles

For similar triangles \(\triangle HBN\sim\triangle LYR\), the ratios of corresponding sides are equal. Assume \(HB\) corresponds to \(LY\) and \(HN\) corresponds to \(LR\). So, \(\frac{HB}{LY}=\frac{HN}{LR}\).
Substitute \(HB = 28\), \(LY=21\), \(HN = 3x - 7\), \(LR=x + 11\) into the proportion: \(\frac{28}{21}=\frac{3x-7}{x + 11}\).

Step2: Simplify the proportion

Simplify \(\frac{28}{21}\) to \(\frac{4}{3}\). The equation becomes \(\frac{4}{3}=\frac{3x-7}{x + 11}\).
Cross - multiply: \(4(x + 11)=3(3x - 7)\).

Step3: Expand the equation

Expand both sides: \(4x+44 = 9x-21\).

Step4: Solve for \(x\)

Move the \(x\) terms to one side and constants to the other side: \(44 + 21=9x-4x\).
\(65 = 5x\).
Divide both sides by \(5\): \(x=\frac{65}{5}=13\).

Answer:

\(13\)