the side AB of parallelogram ABCD is produced both ways to F and G , so that AF = AD and BG = BC. Prove that FD and GC produced intersect at right angles

 

Asked by Paresh | 10th Dec, 2015, 10:58: AM

Expert Answer:

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.
A B C D space i s space a space p a r a l l e log r a m. space B A space i s space p r o d u c e d space t o space F space a n d space A B space i s space p r o d u c e d space t o space G space s u c h space t h a t A F equals A D space a n d space B G equals B C T h u s comma space triangle A D F space a n d space triangle B C G space a r e space i s o s c e l e s space t r i a n g l e s. 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 comma space triangle A F D colon angle A F D equals angle A D F space space open square brackets because triangle A D F space a n d space triangle B C G space a r e space i s o s c e l e s space t r i a n g l e s close square brackets space space space space space space space space space equals angle H D C space space open square brackets because D C parallel to A B rightwards double arrow D C parallel to F G rightwards double arrow c o r r e s p o n d i n g space a n g l e s space a r e space e q u a l space close square brackets C o n s i d e r space t h e space a n g l e space angle A D C space colon space rightwards double arrow angle A D C equals angle H D F minus angle H D C minus angle A F D space space space space space space angle A D C equals 180 degree minus angle H D C minus angle A F D space space space space space space space space open square brackets because angle H D F space i s space a space s t r a i g h t space a n g l e space a n d space angle H D F space equals 180 degree close square brackets space space space angle A D C equals 180 degree minus 2 angle A F D space space space space space space space space space space space space space space space space space space space space space open square brackets because angle A F D equals angle H D C close square brackets  rightwards double arrow angle A B C equals 180 degree minus 2 angle A F D space space space space space space space space space space space space open square brackets because o p p o s i t e space a n g l e s space i n space a space p a r a l l e log r a m space a r e space e q u a l comma space therefore angle A D C equals angle A B C close square brackets rightwards double arrow angle G B C equals space 2 angle A F D space space space space space space space space space space space space space space space space space space space space space open square brackets because angle A B G space i s space a space s t r a i g h t space a n g l e comma angle A B G space equals 180 degree close square brackets space  C o n s i d e r space t h e space triangle B C G colon triangle B C G 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 comma space w i t h space v e r t e x space a t space B. H e n c e space t h e space b a s e space a n g l e s space a r e space e q u a l. B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle G B C plus angle B G C plus angle B C G equals 180 degree rightwards double arrow angle G B C plus 2 angle B G C equals 180 degree space space space space open square brackets because angle B G C equals angle B C G close square brackets space rightwards double arrow space 2 angle A F D space plus 2 angle B G C equals 180 degree space rightwards double arrow space angle A F D space plus angle B G C equals 90 degree space T h u s comma space i n space c o n s i d e r i n g space t h e space t r i a n g l e comma space triangle F G H colon angle H F G equals angle A F D space a n d space angle H G F equals angle B G C therefore space angle A F D space plus angle B G C equals 90 degree rightwards double arrow angle H F G plus angle H G F equals 90 degree rightwards double arrow triangle F G H space i s space a space r i g h t space a n g l e d space t r i a n g l e space r i g h t space a n g l e d space a t space H. space space space space space space space space space space space space space space space space space space space  space space space space space space space space space space space space space space space space space space space space space space space space

Answered by Vimala Ramamurthy | 10th Dec, 2015, 02:35: PM