Question

1. δrst ~ δysz; find yz

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle RST\sim\triangle YSZ\), the ratios of corresponding sides are equal. The ratio of \(SZ\) to \(ST\) is \(\frac{SZ}{ST}=\frac{40 - 5}{40}=\frac{35}{40}=\frac{7}{8}\). Also, for the sides \(YZ\) and \(RT\), we have \(\frac{YZ}{RT}=\frac{SZ}{ST}\). So \(\frac{2x + 2}{3x-7}=\frac{35}{40}\).
Cross - multiply: \(40(2x + 2)=35(3x - 7)\).
Expand: \(80x+80 = 105x-245\).

Step2: Solve for \(x\)

Move \(x\) terms to one side and constants to the other side:
\(80 + 245=105x-80x\).
\(325 = 25x\).
Divide both sides by 25: \(x=\frac{325}{25}=13\).

Step3: Find \(YZ\)

Substitute \(x = 13\) into \(YZ=2x + 2\).
\(YZ=2\times13+2=26 + 2=28\).

Answer:

\(28\)