Question

In patrulaterul ABCD se cunoaste: AB=56cm, BC=CD=42 cm, M€(AD), AM/MD=3/4. Se construieste MN|| DC, N€(AC), NP||AB, P€(BC) si PQ||DC, Q€(BD). Calculați lungimile segmentelor MN, NP, BP, PQ.
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Answer 1

Din AM/MD=3/4⇒AM=3p;MD=4p⇒AD=AM+MD=3p+4p=7p;(p=parte) ⇒  
AM/AD=3p/7p=3/7 . Din MN||DC(prin TFA(Teorema Fundamentala a Asemanarii))⇒ ΔAMN asemenea ΔADC⇒AM/AD=AN/AC=MN/CD=3/7⇒   
MN/42=3/7⇒MN=18 cm. Din AN/AC=3/7⇒AN=3p;AC=7p⇒CN=AC-AN=   
7p-3p=4p⇒CN/CA=4/7. Din NP||AB (prin TFA)⇒ΔCNP asemenea ΔCAB  ⇒ CN/CA=CP/CB=PN/AB=4/7⇒PN/56=4/7⇒PN=32 cm si PC/BC=4/7⇒
PC/42=4/7⇒PC=24 cm⇒ BP=BC-PC=42-24=18 cm.Din CP/CB=4/7⇒  CP=4p;CB=7p⇒BP=CB-CP=7p-4p=3p. Din PQ||CD (prin TFA)⇒ΔBPQ asemenea ΔBCD⇒BP/BC=BQ/BD=PQ/CD=3/7⇒PQ/42=3/7⇒PQ=18 cm