[ Inscribed Quadrilaterals ] [ Area of a Quadrilateral ]

The sum of two opposite sides of an inscribed quadrilateral is 10, and its area is 12. Find the radius of the circle inscribed in this quadrilateral.

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The area of a circumscribed quadrilateral is equal to the product of the semiperimeter and the radius of the inscribed circle.

## Solution

Since the sums of the opposite sides of a circumscribed quadrilateral are equal, its semiperimeter is 10. The area of a circumscribed quadrilateral is equal to the product of the semiperimeter and the radius of the inscribed circle. Therefore, \( r = \frac{12}{10} = \frac{6}{5} \).

![](https://cdn.mathpix.com/cropped/2024_05_06_02603d12f3437ea52c37g-13.jpg?height=373&width=535&top_left_y=88&top_left_x=767)

## Answer

\(\frac{6}{5}\).

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