Question

if diagonal of a cyclic quadrilateral are diameter of the circle through the vertices of the quadrilateral prove it a rectangle

Answer 1

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

angle BAD = 1/2 angle BOD = 180°/2 = 90° (Consider BD as a chord)

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

∠BCD = 180° − 90° = 90°

angle ADC =1/2 angle AOC(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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Answer 2

Hi there!

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For solutions, Refer to the attached picture.
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Let's see some related topics :

⚫ Circle : The collection of all the points, which are at a fixed distance from a fixed point in a plane, is called a circle.

⚫ Radius : A line joining the centre to the Circumference of the circle, is called radius of a circle.

⚫ Secant : A line intersecting a circle at any two points, is called secant.

⚫ Diameter : A chord passing through the point of the circle, is called diameter. It is the longest chord.

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