1.

If the sides of a cyclic quadrilateral are 3,3,4,4, show that a circle be inscribed in it.

Answer»

Solution :Since, length of tangents from EXTERNAL point to a circle are equal.
`THEREFORE` AP=AS………(i)
BP= BQ ……(ii)
CR = CQ ………….(iii)
DR=DS…………..(iv)
`therefore AB+ CD = (AP + PB) (CR + DR)`
`=(AS + BQ) + (CR + DS)
=(AS+DS) + (BQ+CR)`

`rArr AB + CD = AD + BC`.....(1)
`therefore` A circle can be inscribed in a QUADRILATERA, if sides of a quadrilateral satisfies (1)
Sides of quadrilateral are 3,3,4,4.
If we take AB=3, CD=4, BC=3, AD=4
`therefore AB+CD = BC + AD=7`
`therefore` Circle is inscribed in quadrilateral ABCD.
If sum of opposite sides of a quadrilateral is equal. only then a cirlcle can be inscribed in a quadrilateral.


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