Geometry

In the diagram below, EU = 8, UF = 7,GV = 5, VH = 15, and UV= 8.
Find XY.

\(\begin{array}{|rcll|} \hline EU*UF &=& XU*UY \quad | \quad UY=(UV+VY) \\ EU*UF &=& XU*(UV+VY) \\ 8*7 &=& XU*(8+VY) \\ 56 &=& XU*(8+VY) \qquad (1) \\ 56 &=& 8XU+XU*VY \\ XU*VY &=& 56-8XU \qquad (2) \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline GV*VH &=& XV*VY \quad | \quad XV=(XU+UV) \\ GV*VH &=& VY*(XU+UV) \\ 5*15 &=& VY*(8+XU) \\ 75 &=& VY*(8+XU) \qquad (3) \\ 75 &=& 8VY+XU*VY \\ XU*VY &=& 75-8VY \qquad (4) \\ \hline \end{array}\)

\(\begin{array}{|lrcll|} \hline (2)=(4) : & 56-8XU &=& 75-8VY \\ & 8VY &=& 19+8XU \\\\ & VY &=& \dfrac{19+8XU}{8} \qquad (5) \\\\ \hline (2)=(4) : & 56-8XU &=& 75-8VY \\ & 8XU &=& -19+8VY \\\\ & XU &=& \dfrac{-19+8VY}{8} \qquad (6) \\ \hline \end{array}\)

\(\begin{array}{|lrcll|} \hline \text{put}~(6)~\text{into}~(1):& 56 &=& \dfrac{-19+8VY}{8}*(8+VY) \\\\ &8* 56 &=& (-19+8VY)*(8+VY) \\ &448 &=& (-19+8VY)*(8+VY) \\ & 448 &=& -152-19VY+64VY+8VY^2 \\ &8VY^2+45VY-600 &=& 0 \\ \hline &VY &=& \dfrac{-45+\sqrt{21225} }{16} \\ \hline \end{array}\)

\(\begin{array}{|lrcll|} \hline \text{put}~(5)~\text{into}~(3):& 75 &=& \dfrac{19+8XU}{8}*(8+XU) \\\\ & 8*75 &=& (19+8XU)*(8+XU) \\ & 600 &=& (19+8XU)*(8+XU) \\ & 600 &=& 152 +19XU+64XU+8XU^2 \\ & 8XU^2 + 83XU-448 &=& 0 \\ \hline &XU &=& \dfrac{-83+\sqrt{21225} }{16} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline XY &=& UV+XU+VY \\\\ XY &=& 8 + \dfrac{-83+\sqrt{21225} }{16} + \dfrac{-45+\sqrt{21225} }{16} \\\\ XY &=& 8 - \dfrac{83}{16} + \dfrac{\sqrt{21225} }{16} - \dfrac{45}{16} + \dfrac{\sqrt{21225} }{16} \\\\ XY &=& 8 - \dfrac{83+45}{16} + \dfrac{2*\sqrt{21225} }{16} \\\\ XY &=& 8 - 8+ \dfrac{2*\sqrt{21225} }{16} \\\\ XY &=& \dfrac{\sqrt{21225} }{8} \\\\ XY &=& \dfrac{5\sqrt{849} }{8} \\\\ XY &\approx& 18.211 \\ \hline \end{array}\)

In the diagram below, EU = 8, UF = 7,GV = 5, VH = 15, and UV= 8. Find XY.

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UX = n           VY = m

n(8+m) = 56

n = 56 / (8+m)

m(8+n) = 75

m = 75 / (8+n)

m = 75 / [8+56 / (8+m)]

m = 6.293001428

n = 56 / (8 + 6.293001428) = 3.91800143

XY = 18.21100286