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Question Number 154177 by EDWIN88 last updated on 15/Sep/21

Commented by Tawa11 last updated on 15/Sep/21

$$\mathrm{nice}$$

Commented by EDWIN88 last updated on 16/Sep/21

$$Ptolomeo:$$$$\Rightarrow AB.CD+{CD}^2\sin \alpha \sin \theta ={CD}^2\cos \alpha \cos \theta$$$$\Rightarrow AB=CD(\cos \alpha \cos \theta -\sin \alpha \sin \theta )$$$$\Rightarrow AB=CD\cos (\alpha +\theta )$$

Answered by som(math1967) last updated on 15/Sep/21

Commented by som(math1967) last updated on 15/Sep/21

$$\mathit{l}\mathit{e}\mathit{t}\mathit{o}\mathit{i}\mathit{s}\mathit{c}\mathit{e}\mathit{n}\mathit{t}\mathit{r}\mathit{e}\therefore \mathit{C}\mathit{O}=\mathit{A}\mathit{O}=\mathit{B}\mathit{O}=\frac{\mathit{C}\mathit{D}}{2}$$$$\measuredangle \mathit{B}\mathit{C}\mathit{O}=\measuredangle \mathit{O}\mathit{B}\mathit{C}=\mathit{\theta }[\mathit{C}\mathit{O}=\mathit{B}\mathit{O}]$$$$\therefore \measuredangle \mathit{A}\mathit{B}\mathit{O}=\measuredangle \mathit{B}AO=\mathit{\alpha }+\mathit{\theta }$$$$\measuredangle \mathit{A}\mathit{O}\mathit{B}=180-2(\mathit{\alpha }+\mathit{\theta })$$$$\mathit{f}\mathit{r}\mathit{o}\mathit{m}△\mathit{A}\mathit{O}\mathit{B}$$$$\frac{\mathit{A}\mathit{B}}{\mathit{S}\mathit{i}\mathit{n}(180-2\mathit{\alpha }-2\theta )}=\frac{\mathit{A}\mathit{O}}{\mathit{S}\mathit{i}\mathit{n}(\mathit{\alpha }+\mathit{\theta })}$$$$\frac{\mathit{A}\mathit{B}}{\mathit{S}\mathit{i}\mathit{n}2(\mathit{\alpha }+\mathit{\theta })}=\frac{\mathit{A}\mathit{O}}{\mathit{s}\mathit{i}\mathit{n}(\mathit{\alpha }+\mathit{\theta })}$$$$\frac{\mathit{A}B}{2\mathit{s}\mathit{i}\mathit{n}(\mathit{\alpha }+\theta )c\mathit{o}\mathit{s}(\mathit{\alpha }+\mathit{\theta })}=\frac{\mathit{A}O}{\mathit{s}\mathit{i}\mathit{n}(\mathit{\alpha }+\mathit{\theta })}$$$$\mathit{A}\mathit{B}=2\mathit{A}\mathit{O}\mathit{c}\mathit{o}\mathit{s}(\mathit{\alpha }+\mathit{\theta })$$$$\mathit{A}\mathit{B}=\mathit{C}\mathit{D}\mathit{c}\mathit{o}\mathit{s}(\mathit{\alpha }+\mathit{\theta })\left[\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}\right]$$

Commented by mr W last updated on 15/Sep/21

$$OA=OB=\frac{CD}{2}$$$$AB=2\times OB\cos (\theta +\alpha )=CD\cos (\theta +\alpha )$$

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