If Diagonals of a Cyclic Quadrilateral Are Diameters of the Circle Through the Vertices of the Quadrilateral, Prove that It is a Rectangle. - Mathematics

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If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

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Solution

Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O.

`angleBAD=1/2angleBOD=180^@/2=90^@`              (Consider BD as a chord)

∠BCD + ∠BAD = 180°                  (Cyclic quadrilateral)

∠BCD = 180° − 90° = 90°

`angleADC=1/2angleAOC=1/2(180^@) = 90^@`            (Considering AC as a chord)

∠ADC + ∠ABC = 180° (Cyclic quadrilateral)

90° + ∠ABC = 180°

∠ABC = 90°

Each interior angle of a cyclic quadrilateral is of 90°. Hence, it is a rectangle.

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Chapter 10: Circles - Exercise 10.5 [Page 185]

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NCERT Mathematics Class 9
Chapter 10 Circles
Exercise 10.5 | Q 7 | Page 185

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