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ICSE Class 10 Answered

There is no image with me Let you consider it inscribed in circle
Asked by lovemaan5500 | 26 Dec, 2017, 11:58: AM
answered-by-expert 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 | 26 Dec, 2017, 01:51: PM
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