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 how to prove a cyclic quadrilateral as a rectangle

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

ANS :   Let ABCD be a cyclic quadrilateral such that its diagonals AC and BD are the diameters of the circle through the vertices A, B, C, and D.

Ex:3 Cyclic quadrilaterals

 Since AC is a diameter and angle in a semi-circle is a right angle,

 <ABC = 900  and  < ADC = 900

 Similarly, BD is a diameter.

  <DAB = 900  and  <BCD = 900

  Therefore,  < ABC = <ADC = <DAB = <BCD = 900

Thus, ABCD is a rectangle

IF YOU ARE SATISFIED DO GIVE THUMBS UP.

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How do we know that the point of intersection of diagonals of a llgm and the centre of the circle are same??

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Consider a cyclic parallelogram ABCD

<ABC  + <ADC = 180 { Cyclic quadrilateral has opposite sides supplementry}

< ABC  = < ADC  { opposite sides of llgm are equal}

So equating the two we get

<ABC +<ABC= 180

2<ABC= 180

<ABC= 180/2= 90

NOw , we have parallegram ABCD with <ABC = 90

So ABCD is a rectangle as a parallelogram with one angle 90 is a rectangle.

THUS , PROVEN

Hope it was helpful.

Cheers!!

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 is this much sufficient for a 4-mark question?

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 cool

  • -1

 kam ka nahi hain.

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kavi sundar

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not bad but not satisfactory

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