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ICSE Class 10 Answered

In a trapezium ABCD, AB || CD and AD = BC. If P is the point of intersection of the diagonals AC and BD, prove that PC PA = PB PD
Asked by Topperlearning User | 09 Oct, 2017, 01:27: PM
answered-by-expert Expert Answer

begin mathsize 12px style Given space colon space straight A space trapezium space ABCD space in space which space AB space parallel to space DC space and space AD space equals space BC space diagonals space AC space and space BD space intersect space at space straight P To space show colon space PC space cross times space PA space equals space PB space cross times space PD Construction colon space Draw space DM space and space CN space perpendicular space to space AB In space increment space DAM space and space increment space CBN comma DA space equals space CB space space space space left parenthesis given right parenthesis DM equals CN space space space space space left parenthesis perpendicular space distance space between space parallel to space lines space DC space and space AB right parenthesis angle space DMA space equals space angle space CNB space space space space space space space left parenthesis each space 90 degree right parenthesis increment space DAM space approximately equal to space increment space CBN space space space space space space space left parenthesis by space RHS right parenthesis angle space DAM space approximately equal to space angle space CBN space space space space space space space left parenthesis by space cpct right parenthesis rightwards double arrow angle space DAB space equals space angle space CBA space.... left parenthesis 1 right parenthesis angle space CDA space plus space angle space DAB space equals space 180 degree space space space space space space space left parenthesis consecutive space interior space angles space as space DC space parallel to space AB right parenthesis rightwards double arrow angle space CDA space plus space angle space CBA space equals space 180 degree rightwards double arrow ABCD space is space straight a space cyclic space trapezium space as space its space one space pair space of space opposite space angles space are space supplementary. rightwards double arrow AC space and space BD space are space chords space of space straight a space circle space intersecting space at space straight P rightwards double arrow PC space cross times space PA space equals space Pb space cross times space PD end style

Answered by | 09 Oct, 2017, 03:27: PM
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