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ABC is a right angle triangle. D is the point on AB such that AD=3DB. AC=2DB and angle A=90.

Show that sinC=k/square root 20, where k is an interger. Write down the value of k.

$\begin{array}{|rcll|} \hline DB &=& x \\ AD &=& 3x \\ AB &=& AD + DB\\ &=& 3x +x \\ &=& 4x\\\\ AC &=& 2DB \\ &=& 2x \\\\ hypotenuse &=& CB \\ CB^2 &=& AB^2+AC^2 \\ &=& (4x)^2+(2x)^2\\ &=& 16x^2+4x^2\\ &=& 20x^2 \\ CB &=& \sqrt{20x^2} \\ &=& x\cdot \sqrt{20}\\\\ \sin(C) &=& \frac {AB}{CB} \\ &=& \frac {4x}{x\cdot \sqrt{20}} \\ &=& \frac {\mathbf{4} } { \sqrt{20} } \\ \text{So the value of k is }\mathbf{4} \\ \hline \end{array}$