As shown in the figure, an irregular quadrilateral is inscribed in a semicircle with its base being the diameter of the semicircle. Find the radius r of the semicircle.

Guest Dec 19, 2019

#1**+2 **

**As shown in the figure, an irregular quadrilateral is inscribed in a semicircle with its base being the diameter of the semicircle. **

**Find the radius r of the semicircle.**

\(\begin{array}{|lrcll|} \hline (1) & 180^\circ &=& 4A+2B \quad | \quad :2 \\ & 90^\circ &=& 2A+B \\ & \mathbf{B} &=& \mathbf{90^\circ -2A} \\\\ (2) & \sin(B) &=& \dfrac{7}{r} \quad | \quad \mathbf{B=90^\circ -2A} \\\\ & \sin(90^\circ -2A) &=& \dfrac{7}{r} \\\\ & \cos(2A) &=& \dfrac{7}{r} \quad | \quad \cos(2A) = 1-2\sin^2(A) \\\\ & 1-2\sin^2(A) &=& \dfrac{7}{r} \quad | \quad \mathbf{\sin(A)=\dfrac{3}{r} } \\\\ & 1-2\left(\dfrac{3}{r}\right)^2 &=& \dfrac{7}{r} \\\\ & 1-\dfrac{2*9}{r^2} &=& \dfrac{7}{r} \\\\ & 1-\dfrac{18}{r^2} &=& \dfrac{7}{r} \quad | \quad * r^2 \\\\ & r^2-18 &=& 7r \\ & r^2-7r-18 &=& 0 \\ &(r-9)(r+2) &=& 0 \\ &\mathbf{r} &=& \mathbf{9} \\ \hline \end{array}\)

heureka Dec 19, 2019