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# geometry

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A circle is inscribed in a quarter-circle sector. It is tangent to the arc of the sector and the two perpendicular radii of the sector. What is the ratio of the area of the inscribed circle to the area of the sector?

Jul 20, 2020

### 2+0 Answers

#1
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A circle is inscribed in a quarter-circle sector. It is tangent to the arc of the sector and the two perpendicular radii of the sector. What is the ratio of the area of the inscribed circle to the area of the sector?

Hello Guest!

$$a=r+r\sqrt{2}\\ a=r(1+\sqrt{2})$$          Radius of the sector

$$A_{sec}=\pi r^2(1+\sqrt{2})^2/4$$

$$A_{insc}=\pi r^2$$

$$\frac{A_{insc}}{A_{sec}}=\frac{4}{(1+\sqrt{2})^2}$$

$$\frac{A_{insc}}{A_{sec}}=\frac{2}{1+\sqrt{2}}=1:2,414$$

!

Jul 20, 2020
edited by asinus  Jul 20, 2020
#2
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1 : 2.414        is the ratio of their radii

Ratio:>>         Ainsc  :  Asec  = 11 : 16   or  11/16

Guest Jul 20, 2020