Question

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

Answer 1

Given,
ABCD is a cyclic quadrilateral and AC and BD are the diameters of the circle.
To prove: ABCD is a rectangle.
Proof: To prove that ABCD is a rectangle, we only need to prove that its all angles are 90 degrees.

In semi-ciecle ADC,
angle ADC = 90 degrees (since, the line segment from the two ends of the diameter of a semi-circle makes 90 degrees on the circumference.)
Similarly, we can prove that angle ABC, angle BCD and angle DAB is equal to 90 degrees.

Therefore, ABCD is a rectangle.
Hence, Proved.

Answer 2

Hi there!

_______________________

For solutions, 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 !

☺️❤️☺️