Question

if digonal of cyclic quadrilateral are diameters of the cricle though the vertices of the quadrilateral prove that it is a rectngle

Answer 1

Let diagonals AC and BD of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral ABCD.

From Figure,

OA = OB = OC = OD (Since all radius of a circle are equal)

=> OA = OC = 1/2 * AC ..........1

and OB = OD = 1/2 * BD ..........2

From equation 1 and 2, we get

1/2 * AC = 1/2 * BD

=> AC = BD

So diagonals AC and BD of the quadrilateral ABCD are equal and bisect each other.

Answer 2

Hello mate =_=

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Solution:

It is given that ABCD is a cyclic quadrilateral. Also, AC and BD are two diameters of circle having centre O.

We need to prove that ABCD is a rectangle.

BD is a diameter which means that ∠BCD=∠DAB=90°           ......... (1)

(Angle in a semi-circle is equal to 90°)

Similarly, AC is a diagonals which means that ∠ABC=∠ADC=90°  .....(2)

(Angle in a semi-circle is equal to 90°)

From (1) and (2), we can notice that opposite angles of quadrilateral ABCD are equal which makes it a parallelogram.

Also, all the corner angles are equal to 90° which makes it a rectangle.

I hope, this will help you.

Thank you______❤

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