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Ex 10.5, 7 - If diagonals of cyclic quadrilateral are diameters - Ex 10.5

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Slide13.JPG

  1. Chapter 10 Class 9 Circles
  2. Serial order wise
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Ex 10.5, 7 If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. Given: Let ABCD be a cyclic quadrilateral where diagonals AC & BD are diameters, To prove: ABCD is a rectangle Proof: A rectangle is a parallelogram with one angle 90° So, we first prove ABCD is a parallelogram, then one angle 90 Since BD is the diameter Arc BAD is a semicircle, So, ∠BAD = 90° Also, ABCD is a cyclic quadrilateral Sum of Opposite angles of cyclic quadrilateral is 180° So, in quadrilateral ABCD ∠A = ∠B = ∠C = ∠D = 90° Since ∠A = ∠C & ∠B = ∠D , i.e. both pairs of opposite angles are equal ABCD is a parallelogram Also, all angles are 90° So, ABCD is a parallelogram with one angle 90 ° Therefore, ABCD is a rectangle Hence proved

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