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Asked by lovemaan5500 | 26th Dec, 2017, 11:58: AM

Expert Answer:

In a trapezium ABCD .AB //CD and AD=BC .If P is point of intersection of diagonals AC and BD prove that PA×PC=PB×PD
 
begin mathsize 16px style Construction colon space Draw space DE perpendicular AB space and space CF perpendicular AB
In space triangle DEA space and space triangle CFB comma
AD equals BC space space space space left parenthesis given right parenthesis
angle AEB equals angle CFB space space space left parenthesis Each space 90 degree right parenthesis
DE equals CF space space left parenthesis distance space between space two space parallels right parenthesis
rightwards double arrow triangle DEA approximately equal to triangle CFB space left parenthesis RHS space criterion right parenthesis
rightwards double arrow angle DAE equals angle CBF space space space space space left parenthesis cpct right parenthesis
Now comma space angle straight D plus angle straight B equals angle ADC plus angle CBA
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 equals angle ADC plus angle CBF
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 equals angle ADC plus angle DAE
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 equals 180 degree space space space left parenthesis sum space of space co minus interior space angles right parenthesis
rightwards double arrow Opposite space angles space of space trapezium thin space ABCD space are space supplementary.
rightwards double arrow ABCD space is space straight a space cyclic space quadrilateral.
Thus comma space AC space and space BD space are space two space chords space of space the space circle space circumscribing space the space trapezium
such space that space they space intersect space at space straight P.
Hence comma space PA cross times PC equals PB cross times PD end style
 

Answered by Rashmi Khot | 26th Dec, 2017, 01:51: PM

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