Question

solve for z. z =

Explanation:

Step1: Use the geometric mean theorem

In a right - triangle, if a height is drawn to the hypotenuse, then the length of the height is the geometric mean of the lengths of the segments of the hypotenuse. Also, \(\frac{VW}{WX}=\frac{VY}{YZ}\). Given \(VW = 9\), \(WY=27\), \(WX = 5\), and \(YZ=z\). First, note that \(\triangle VWX\sim\triangle VYZ\) (by AA similarity, as \(\angle V=\angle V\) and \(\angle VWX=\angle VYZ = 90^{\circ}\)).
The ratio of the sides of similar triangles gives \(\frac{VW}{VY}=\frac{WX}{YZ}\). Since \(VY=VW + WY=9 + 27=36\).

Step2: Substitute values into the proportion

Substitute \(VW = 9\), \(VY = 36\), and \(WX = 5\) into \(\frac{VW}{VY}=\frac{WX}{YZ}\). We get \(\frac{9}{36}=\frac{5}{z}\).
Cross - multiply: \(9z=36\times5\).

Step3: Solve for \(z\)

\(9z = 180\). Divide both sides by 9: \(z=\frac{180}{9}=20\).

Answer:

\(20\)