Question

Fie punctele A, B, C colineare in aceasta ordine si punctele M, N, P mijloacele segmentelor (AB) , (BC) . Daca MB=7 cm si MC= 15 cm, atunci determinați lungimea segmentelor [AB] ; [BC]; [AC]; [MP] si [NP]

Answer 1

|.........||......|...||.........||.......|A       M     P  B        N      C    
AB=2*AM=2*5=10
BC=MC-MPBC=11-5BC=6
AC=AB+BC=10+6=16
MP=AC/2= 16/2=8
NC=BN=BC/2=6/2=3
AP=PC=AC/2=16/2=8
NP=8-5NP=3A            M            B                             P                             C
|_______!_______|_______________!_______________|

AP = 16   => AB + BP = 16  =>  AB + BC/2 = 16
MC = 20  => MB + BC = 20 =>  AB/2 + BC  = 20
Rezolvam sistemul de ecuatii:
AB + BC/2 = 16        | × 2
AB/2 + BC  = 20       | × 2
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2AB + BC = 32    => Substitutia:  BC = 32 - 2AB
AB + 2BC = 40   
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AB + 2(32 - 2AB) = 40
AB + 64 -4AB = 40
3AB = 64 - 40
3AB  = 24
AB = 24 / 3 = 8 
---
BC = 32 - 2AB = 32 - 2×8 = 32 - 16 = 16
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a)  MP = MB + BP = AB/2 + BC/2 = 8/2 + 16/2 = 4 + 8 = 12

b) AB = 8    

c)  BC = 16

d)  AC = AB + BC = 8 + 16 = 24