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Question Number 209783 by Ismoiljon_008 last updated on 21/Jul/24

Answered by mr W last updated on 21/Jul/24

Commented by mr W last updated on 21/Jul/24

$$AC=\sqrt{{12}^2+{16}^2}=20\Rightarrow radius=10$$$$AB=12\times \frac{12}{20}=\frac{36}{5}$$$$BC=16\times \frac{16}{20}=\frac{64}{5}$$$$OB=10-\frac{36}{5}=\frac{14}{5}$$$$OP=\frac{14}{5}+r=\sqrt{{(10-r)}^2-r^2}$$$$\Rightarrow 14+5r=10\sqrt{5(5-r)}$$$$\Rightarrow 25r^2+640r-2304=0$$$$\Rightarrow r=\frac{-320+400}{25}=\frac{16}{5}$$$$OQ=R-\frac{14}{5}=\sqrt{{(10-R)}^2-R^2}$$$$\Rightarrow 5R-14=10\sqrt{5(5-R)}$$$$\Rightarrow 25R^2+360R-2304=0$$$$\Rightarrow R=\frac{-180+300}{25}=\frac{24}{5}$$$$PQ=r+R=\frac{16}{5}+\frac{24}{5}=8✓$$

Commented by Ismoiljon_008 last updated on 21/Jul/24

$$thankyouverymuch$$

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