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Points A, B, and C are on sides , , and of rectangle WXYZ as shown below such that XA = 4, YB = 18, ​, and YC = 2XC. Find AB.

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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.

Apr 6, 2020

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deledet

Apr 6, 2020
edited by Guest  Apr 6, 2020
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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 $$AB$$.

$$\begin{array}{|rcll|} \hline \tan(A) = \dfrac{4}{XC} &=& \dfrac{2XC}{18} \\\\ \dfrac{4}{XC} &=& \dfrac{2XC}{18} \\\\ \dfrac{4}{XC} &=& \dfrac{XC}{9} \\\\ 36 &=& XC^2 \\\\ 6 &=& XC \\ \mathbf{XC} &=& \mathbf{6} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline x^2 &=& (3XC)^2 + 14^2 \\ x^2 &=& (3*6)^2 + 14^2 \\ x^2 &=& 18^2 + 14^2 \\ x^2 &=& 324 + 196 \\ x^2 &=& 520 \\ x^2 &=& 4*130 \\ x &=& 2\sqrt{130} \\ \mathbf{x} &=& \mathbf{22.8035085020} \\ \hline \end{array}$$

$$\mathbf{AB \approx 22.8}$$

Apr 6, 2020