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Standard X
Mathematics
Locus of the Points Equidistant From Two Given Points
The given fig...
Question

The given figure shows a triangle ABC in whcih AD bisects angle BAC. EG is perpendicular bisector of side AB which intersects AD at point F.

Prove that:

(i) F is equidistant from A and B.

(ii) F is equidistant from AB and AC.

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Solution

(i)

C o n s t r u c t i o n colon space space J o i n space space F B space a n d space space F C space P r o o f colon space space I n space increment A F E space a n d space increment F B E comma space A E space equals space E B space space left parenthesis E space i s space space t h e space m i d space minus space p o i n t space space o f space space A B right parenthesis angle space F E A space equals space space angle space F E B space left parenthesis E a c h space space equals space 90 degree right parenthesis space F E space equals space F E space left parenthesis C o m m o n right parenthesis therefore space B y space space s i d e space space A n g l e space space s i d e space space c r i t e r i o n space space o f space c o n g r u e n c e comma space increment A F E space space approximately equal to space increment F B E space space space left parenthesis S A S space P o s t u l a t e right parenthesis T h e space c o r r e s p o n d i n g space space p a r t s space space o f space t h e space c o n g r u e n t space space t r i a n g l e s space space a r e space c o n g r u e n t. space therefore space A F space equals space F B space left parenthesis C P C T right parenthesis H e n c e comma space F space i s space e q u i d i s tan t space f r o m space A space a n d space B.
(ii)

C o n s t r u c t i o n colon space space D r a w space L F space space perpendicular space A C P r o o f colon space space I n space increment space A F L space a n d space increment A F E comma angle space F E A space equals space space angle space F L A space space left parenthesis E a c h space space equals space 90 degree right parenthesis angle space L A F space equals space F A E space space left parenthesis A D space space b i space s e c t s space space angle space B A C right parenthesis A F space equals space A F space left parenthesis c o m m o n right parenthesis therefore space B y space space a n g l e space space � space A n g l e space space s i d e space c r i t e r i o n space space o f space c o n g r u e n c e comma space increment A F L space space approximately equal to space A F E space space left parenthesis A A S space space p o s t u l a t e right parenthesis T h e space space c o r r e s p o n d i n g space space p a r t s space space o f space t h e space c o n g r u e n t space space t r i a n g l e s space space a r e space c o n g r u e n t. space therefore space F E space equals space F L space left parenthesis C P C T right parenthesis H e n c e comma space F space i s space e q u i d i s tan t space f r o m space A B space a n d space A C.


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