Question

in the following triangle, find length wy. round to the nearest tenth.

Explanation:

Step1: Apply the geometric mean theorem (altitude-on-hypotenuse theorem)

In a right - triangle \( \triangle XYW\) with altitude \( YZ\), we have the relationship \( YZ^{2}=XZ\times ZW\). First, find \( XZ\) using the Pythagorean theorem in \( \triangle XYZ\): \( XZ=\sqrt{XY^{2}-YZ^{2}}=\sqrt{7^{2}-5.7^{2}}=\sqrt{49 - 32.49}=\sqrt{16.51}\approx4\) (already given). Then, from \( YZ^{2}=XZ\times ZW\), we can rewrite it for \( ZW\) as \( ZW=\frac{YZ^{2}}{XZ}\). Substituting \( YZ = 5.7\) and \( XZ=4\), we get \( ZW=\frac{5.7^{2}}{4}=\frac{32.49}{4}=8.1225\).

Step2: Use the Pythagorean theorem in \( \triangle YZW\)

In right - triangle \( \triangle YZW\), by the Pythagorean theorem \( WY=\sqrt{YZ^{2}+ZW^{2}}\). Since \( YZ = 5.7\) and \( ZW = 8.1225\), then \( WY=\sqrt{5.7^{2}+8.1225^{2}}=\sqrt{32.49+66.0}\) (approximate \( 8.1225^{2}\approx66.0\)) \(=\sqrt{98.49}\approx9.9\).

Another way:

Step1: Use the geometric mean relationship for the segments of the hypotenuse and the legs

In a right - triangle \( \triangle XYW\) with altitude \( YZ\), the relationship \( XY^{2}=XZ\times XW\) and \( WY^{2}=ZW\times XW\). Also, \( XW=XZ + ZW\). From \( YZ^{2}=XZ\times ZW\) (where \( YZ = 5.7\), \( XZ = 4\)), we know \( ZW=\frac{5.7^{2}}{4}=8.1225\). And \( XW=4 + 8.1225=12.1225\).

Step2: Apply the formula \( WY^{2}=ZW\times XW\)

Substitute \( ZW = 8.1225\) and \( XW=12.1225\) into \( WY^{2}=ZW\times XW\). Then \( WY=\sqrt{8.1225\times12.1225}=\sqrt{98.49}\approx9.9\)

Answer:

\(9.9\)