ABCD is a parallelogram, P is any point on AC. Through P, MN is drawn parallel to BA cutting BC in

Nickolas Taylor 2022-07-14 Answered
ABCD is a parallelogram, P is any point on AC. Through P, MN is drawn parallel to BA cutting BC in M and AD in N. SR is drawn parallel to BC cutting BA in S and CD in R. Show that [ASN]+[AMR]=[ABD] (where [ . ] denotes the area of the rectilinear figure).

My attempt : used base division method to find the area of ASN but I get extra variables which is tough...
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Answers (2)

Maggie Bowman
Answered 2022-07-15 Author has 14 answers
[ . ] represents area
Draw D P and B P and using the properties of parallelogram (diagonal bisects the area)
[ P M R ] = [ C M R ]
[ A S N ] = [ P S N ]
[ D P R ] = [ A P R ] (same base, equal height) and [ D P R ] = [ N P D ]
[ B P M ] = [ A P M ] (same base, equal height) and [ B P M ] = [ S P B ]
[ P M R ] + [ A S N ] + [ A P R ] = [ A P M ] = [ A B C D ] 2 = [ A B D ]
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Keenan Santos
Answered 2022-07-16 Author has 1 answers
Note [ A S N ] = 1 2 [ A S P N ], [ A B D ] = 1 2 [ A B C D ].
For A M R,
[ A M R ] = [ A B C D ] [ A B M ] [ C R M ] [ A R D ]
= [ A B C D ] 1 2 [ A B M N ] 1 2 [ C R P M ] 1 2 [ A S R D ]
= [ A B C D ] 1 2 ( [ A B M N ] + [ C R P M ] + [ A S R D ] )
= [ A B C D ] 1 2 ( [ A B C D ] + [ A S P N ] )
= 1 2 [ A B C D ] 1 2 [ A S P N ]
= [ A B D ] [ A S N ]
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