Question

1. if $\triangle hbn sim \triangle lyr$, find the value of $x$. 4 points

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle HBN\sim\triangle LYR\), the ratios of corresponding sides are equal. So, \(\frac{HB}{LY}=\frac{HN}{LR}\).

Step2: Substitute the given values

We have \(HB = 28\), \(LY=21\), \(HN = 3x - 7\), and \(LR=x + 11\). Then \(\frac{28}{21}=\frac{3x - 7}{x + 11}\).

Step3: Simplify the left - hand side ratio

\(\frac{28}{21}=\frac{4}{3}\), so the equation becomes \(\frac{4}{3}=\frac{3x - 7}{x + 11}\).

Step4: Cross - multiply

\(4(x + 11)=3(3x - 7)\).

Step5: Expand both sides

\(4x+44 = 9x-21\).

Step6: Solve for \(x\)

Subtract \(4x\) from both sides: \(44=9x - 4x-21\), which simplifies to \(44 = 5x-21\). Add \(21\) to both sides: \(44 + 21=5x\), so \(65 = 5x\). Divide both sides by \(5\): \(x = 13\).

Answer:

\(x = 13\)