AC and BD are chords of a circle which bisect each other. Prove that

(i) AC and BD are diameters of the circle and

(ii) ABCD is a rectangle.



Asked by Topperlearning User | 4th Oct, 2017, 01:42: PM

Expert Answer:

begin mathsize 12px style Since comma space AC space and space BD space bisect space each space other.
AC space and space BD space are space diagonals space of space parallelogram space ABCD.
ABCD space is space parallelogram.
angle space straight A space equals space angle space straight C space and space angle space straight B space equals space angle space straight D
ABCD space is space also space straight a space cyclic space quadrilateral.
angle space straight A space plus space angle space straight C space equals space 180 degree space and space angle space straight B space plus space angle space straight D space equals space space 180 degree space
rightwards double arrow space 2 angle space straight A space equals space space 180 degree space and space 2 angle space straight B space equals space 180 degree
rightwards double arrow angle space straight A space equals space 90 degree space space space and space space angle space straight B space equals space space 90 degree space
rightwards double arrow angle space straight A space equals space angle space straight C space equals space angle space straight B space equals space angle space straight D space equals space 90 degree space
rightwards double arrow space ABCD space is space straight a space rectangle. space and space
AC space and space BD space are space the space diameters space of space the space circle space space space space space space left parenthesis space angle space in space straight a space semicircle space is space 90 degree right parenthesis
Hence comma space proved space left parenthesis straight i right parenthesis space AC space and space BD space are space the space diameters space of space the space circle
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis ii right parenthesis space ABCD space is space straight a space rectangle. end style

Answered by  | 4th Oct, 2017, 03:42: PM