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CBSE Class 10 Answered

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joinign the end points of an arc.
Asked by amoghknightoo7 | 11 Mar, 2016, 11:08: AM
answered-by-expert Expert Answer
 
I n space t h e space f i g u r e comma space straight C space i s space t h e space m i d p o i n t space o f space t h e space m i n o r space a r c space P Q comma straight O space i s space t h e space c e n t r e space o f space t h e space c i r c l e space a n d space A B space i s space t a n g e n t space t o space t h e space c i r c l e space t h r o u g h space p o i n t space straight C. W e space space h a v e space t o space s h o w space t h e space t a n g e n t space d r a w n space a t space t h e space m i d p o i n t space o f space t h e space a r s space P Q space o f space straight a space c i r c l e space i s space p a r a l l e l space t o space t h e space c h o r d space j o i n i n g space t h e space e n d space p o i n t s space o f space t h e space a r c space P Q. straight i. straight e. space W e space w i l l space s h o w space P Q space vertical line vertical line space A B. I t space i s space g i e n space t h a t space straight C space i s space m i d p o i n t space o f space t h e space a r c space P Q. S o comma space a r c space P C space equals space a r c space C Q rightwards double arrow space P C equals C Q T h i s space s h o w s space t h a t space increment P Q C space i s space a n space i s o s c e l e s space t r i a n g l e. T h u s comma space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space s i d e space P Q space o f space increment P Q C space p a s s e s space t h r o u g h space v e r t e x space straight C. T h e space p e r p e n d i c u l a r space b i s e c t o r space o f space straight a space c h o r d space p a s s e s space t h r o u g h space t h e space c e n t r e space o f space t h e space c i r c l e. S o space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space P Q space space p a s s e s space t h r o u g h space t h e space c e n t r e space straight O space o f space t h e space c i r c l e. T h u s space p e r p e n d i c u l a r space b i s e c t o r space o f space P Q space p a s s e s space t h r o u g h space t h e space p o i n t s space straight O space a n d space straight C. rightwards double arrow P Q perpendicular O C A B space i s space t h e space t a n g e n t space t o space t h e space c i r c l e space t h r o u g h space t h e space p o i n t space straight C space o n space t h e space c i r c l e. rightwards double arrow A B perpendicular O C T h e space c h o r d space P Q space a n d space t h e space t a n g e n t space P Q space o f space t h e space c i r c l e space a r e space p e r p e n d i c u l a r space t o space t h e space s a m e space l i n e space O C. therefore P Q space vertical line vertical line space A B
Answered by Rashmi Khot | 11 Mar, 2016, 01:03: PM

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