Points A, B, and C are on sides \(\overline{WX}\), \(\overline{YZ}\), and \(\overline{XY}\) of rectangle WXYZ as shown below such that XA = 4, YB = 18, \(\angle ACB= 90^\circ\), and YC = 2XC. Find AB.
+26396 Points \(A\), \(B\), and \(C\) are on sides\( \overline{WX}\), \(\overline{YZ}\), and \(\overline{XY}\) of rectangle \(WXYZ\) as shown below such that
\(XA = 4\), \(YB = 18\), \(\angle ACB= 90^\circ\), and \(YC = 2XC\).
Find \(AB\).
\(\begin{array}{|rcll|} \hline \tan(A) = \dfrac{4}{XC} &=& \dfrac{2XC}{18} \\\\ \dfrac{4}{XC} &=& \dfrac{2XC}{18} \\\\ \dfrac{4}{XC} &=& \dfrac{XC}{9} \\\\ 36 &=& XC^2 \\\\ 6 &=& XC \\ \mathbf{XC} &=& \mathbf{6} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline x^2 &=& (3XC)^2 + 14^2 \\ x^2 &=& (3*6)^2 + 14^2 \\ x^2 &=& 18^2 + 14^2 \\ x^2 &=& 324 + 196 \\ x^2 &=& 520 \\ x^2 &=& 4*130 \\ x &=& 2\sqrt{130} \\ \mathbf{x} &=& \mathbf{22.8035085020} \\ \hline \end{array}\)
\(\mathbf{AB \approx 22.8}\)
