Question
Fri March 20, 2015 By: Sriradha Gupta

In a circle with centre of radius 17cm, chord AB and chord CD are drawn parallel such that the sum of their distances from the centre O is 23cm. If AB=16 cm, find the area of trapezium ABCD.(Join AD and BC).

Expert Reply
Vimala Ramamurthy
Sun March 22, 2015
C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
G i v e n space t h a t space t h e space r a d i u s space o f space t h e space c i r c l e space i s space 17 space c m. rightwards double arrow O A equals O B equals O C equals O D equals 17 space c m A B space a n d space C D space a r e space t w o space c h o r d s space o f space t h e space c i r c l e. A B equals 16 space c m S i n c e space t h e space p e r p e n d i c u l a r space l i n e space j o i n i n g space t h e space c e n t r e space a n d space c h o r d space b i s e c t s space t h e space c h o r d comma space A M equals B M space a n d space C N equals D N rightwards double arrow A M equals B M equals 16 over 2 equals 8 space c m A n d triangle O A M comma space triangle O B M comma space triangle O C N space a n d space triangle O D N space a r e space r i g h t space t r i a n g l e s. H e n c e space b y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space i n space triangle O A M comma space w e space h a v e comma O A squared equals A M squared plus O M squared rightwards double arrow 17 squared equals 8 squared plus O M squared rightwards double arrow 289 equals 64 plus O M squared rightwards double arrow O M squared equals 289 minus 64 rightwards double arrow O M squared equals 225 rightwards double arrow O M equals square root of 225 rightwards double arrow O M equals 15 space c m G i v e n space t h a t space M N equals 23 space c m space a n d space M N equals M O plus O N comma O N equals M N minus M O equals 23 minus 15 equals 8 space c m N o w space c o n s i d e r space t h e space t r i a n g l e space triangle O C N B y space P y t h a g o r a s space T h e o r e m comma space w e space h a v e O C squared equals C N squared plus O N squared rightwards double arrow 17 squared equals C N squared plus 8 squared rightwards double arrow 289 equals C N squared plus 64 rightwards double arrow C N squared equals 289 minus 64 rightwards double arrow C N squared equals 225 rightwards double arrow C N equals square root of 225 rightwards double arrow C N equals 15 space c m C D equals C N plus N D equals 2 C N equals 2 cross times 15 equals 30 space c m N o w space t h e space l e n g t h s space o f space t h e space p a r a l l e l space s i d e s space A B space a n d space C D space o f space t h e space t r a p e z i u m space a r e space 16 space c m a space n d space 30 space c m space r e s p e c t i v e l y. A r e a space o f space t h e space t r a p e z i u m equals 1 half cross times s u m space o f space t w o space p a r a l l e l space s i d e s cross times h e i g h t space o f space t h e space t r a p e z i u m equals 1 half cross times open parentheses 16 plus 30 close parentheses cross times 23 equals 529 space c m squared
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