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Question Number 189921 by cortano12 last updated on 26/Mar/23

Commented by nikif99 last updated on 25/Mar/23

Commented by cortano12 last updated on 26/Mar/23

$$\mathrm{i}\mathrm{have}\mathrm{corrected}\mathrm{the}\mathrm{question}$$

Answered by nikif99 last updated on 25/Mar/23

$$PD\mathrm{\perp }BC\Rightarrow CD=7\Rightarrow CP=d=\frac{7}{\cos \theta }\left(1\right)$$$$△ABC:\cos C=\frac{{AC}^2+{BC}^2-{AB}^2}{2\cdot AC\cdot BC}=0.6\Rightarrow \sin C=0.8$$$$\measuredangle \omega =360-\measuredangle CPB-\measuredangle BPA=$$$$360-(180-2\theta )-[180-(\measuredangle A-\theta )-(\measuredangle B-\theta )]=$$$$360-180+2\theta -180+\measuredangle A-\theta +\measuredangle B-\theta \Rightarrow$$$$\measuredangle \omega =\measuredangle A+\measuredangle B=180-\measuredangle C$$$$△ACP:\frac{d}{\sin \theta }=\frac{AC}{\sin \omega }=\frac{15}{\sin \measuredangle C}=\frac{15}{0.8}\Rightarrow$$$$\frac{d}{\sin \theta }=18.75\left(2\right)$$$$\left(1\right)\left(2\right)\Rightarrow \frac{7}{\cos \theta \sin \theta }=18.75\Rightarrow \frac{14}{2\cos \theta \sin \theta }=18.75\Rightarrow$$$$\sin \left(2\theta \right)=\frac{14}{18.75}\Rightarrow 2\theta =48.302$$$$and\theta =24.151\Rightarrow \tan \theta =0.448395=\frac{a}{b}$$$$Thenbycomputeraidedprogram,$$$$reacheda=908,b=2025anda+b=2933$$

Answered by mr W last updated on 18/Nov/23

$$s=\frac{13+14+15}{2}=21$$$$\mathrm{\Delta }=\sqrt{21\times (21-13)(21-14)(21-15})=84$$$$\tan \theta =\frac{4\times 84}{{13}^2+{14}^2+{15}^2}=\frac{168}{295}=\frac{a}{b}$$$$\Rightarrow a+b=168+295=463✓$$

Commented by mr W last updated on 18/Nov/23

$$\text{You can't use 'macro parameter character \#' in math mode}$$

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