Question

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

Answer 1

Hello,
let's look at the figure.
Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O.

$\widehat{BAD}=\frac{1}{2}\widehat{BOD}=\frac{180}{2}=90^{\circ}$

Consider BD as a chord 
$\widehat{BCD}+\widehat{BAD}=180^{\circ}$(Cyclic quadrilateral)
$\widehat{BCD}=180-90=90^{\circ}$

Considering AC as a chord
$\widehat{ADC}+\widehat{ABC}=180^{\circ}$(  (Cyclic quadrilateral)
$90^{\circ}+\widehat{ABC}=180^{\circ}$
$\widehat{ABC}=90^{\circ}$

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

bye :-)

Answer 2

Hi there!

_______________________

For solutions to the question, Refer to the attached picture.

Regrets for handwriting _/\_

_______________________

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.

_______________________

Thanks for the question !

☺️❤️☺️