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CBSE Class 9 Answered

Prove that the straight line joining the mid - points of the diagonals of a trapezium is parallel to the parallel sides and is equal to half their difference.
Asked by dimplesharma0810 | 26 Feb, 2016, 06:30: PM
answered-by-expert Expert Answer
 
begin mathsize 14px style G i v e n colon A thin space t r a p e z i u m space A B C D space i n space w h i c h space A B space parallel to D C space a n d space P space a n d space Q space a r e space t h e space m i d space p o i n t s space o f space d i a g o n a l s space A C space a n d space B D space r e s p e c t i v e l y. T o space p r o v e colon 1 right parenthesis P Q space parallel to space A B space o r space D C 2 right parenthesis space P Q equals 1 half left parenthesis A B minus D C right parenthesis P r o o f colon S i n c e space A B space parallel to D C space a n d space t r a n s v e r s a l space A C space c u t s space t h e m space a t space A space a n d space C space r e s p e c t i v e l y comma angle 1 equals angle 2 space space space space space.... left parenthesis 1 right parenthesis left parenthesis A l t e r n a t e space a n g l e s right parenthesis N o w comma space i n space t r a i n g l e s space A P R space a n d space D P C comma angle 1 equals angle 2 space space space space space.... left parenthesis F r o m space left parenthesis 1 right parenthesis right parenthesis A P equals C P space space space space space... left parenthesis P space i s space m i d minus p o i n t space o f space A C right parenthesis angle 3 equals angle 4 space space space space space... left parenthesis v e r t i c a l l y space o p p o s i t e space a n g l e s right parenthesis rightwards double arrow triangle A P R space approximately equal to triangle D P C space b y space A S A thin space c r i t e r i o n rightwards double arrow A R equals D C space a n d space P R equals D P space space space space space... left parenthesis C P C T right parenthesis I n space triangle D R B comma space P space a n d space Q space a r e space m i d minus p o i n t s space o f space s i d e s space D R space a n d space D B space r e s p e c t i v e l y. rightwards double arrow P Q space parallel to R B rightwards double arrow P Q parallel to A B rightwards double arrow P Q space parallel to A B space a n d space D C A g a i a n comma space P space a n d space Q space a r e space m i d minus p o i n t s space o f space s i d e s space D R space a n d space R B space r e s p e c t i v e l y space i n space triangle D R B. rightwards double arrow P Q equals 1 half R B rightwards double arrow P Q equals 1 half left parenthesis A B minus A R right parenthesis rightwards double arrow P Q equals 1 half left parenthesis A B minus D C right parenthesis end style
Answered by Rashmi Khot | 26 Feb, 2016, 07:09: PM

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