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If the Sides of a Parallelogram Touch a Circle (Refer Figure of Q. 7), Prove that the Parallelogram is a Rhombus - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

Question

If the sides of a parallelogram touch a circle (refer figure of Q. 7), Prove that the parallelogram is a rhombus

Solution

From A, AP and AS are tangents to the circle.
Therefore, AP = AS.......(i)
Similarly, we can prove that:
BP = BQ .........(ii)
CR = CQ .........(iii)
DR = DS .........(iv)
Adding,
AP + BP + CR + DR = AS + DS + BQ + CQ
AB + CD = AD + BC
Hence, AB + CD = AD + BC
But AB = CD and BC = AD.......(v) Opposite sides of a ||gm
Therefore, AB + AB = BC + BC
2AB = 2 BC
AB = BC ........(vi)

From (v) and (vi)
AB = BC = CD = DA
Hence, ABCD is a rhombus

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Solution If the Sides of a Parallelogram Touch a Circle (Refer Figure of Q. 7), Prove that the Parallelogram is a Rhombus Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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