Question

Problema 17.........

Answer 1

fiind patrat toate laturile sunt egale adica
$AB=BC=CD=AD=l=24$
Atunci relatiile de mai sus devin
$AM=BM=\frac{l}{2}$
$BN=\frac{l}{3}$ si $CN=BC-BN=l-\frac{l}{3}=\frac{2l}{3}$
$CP=\frac{l}{4}$ si $DP=CD-CP=l-\frac{l}{4}=\frac{3l}{4}$
$DQ=\frac{l}{6}$ si $AQ=AD-DQ=l-\frac{l}{6}=\frac{5l}{6}$
Atunci putem sa calculam ariile triunghiurilor dreptunghice formate din unghiurile drepte ale patratului cu ipotenuzele reprezentand laturile patrulaterului MNPQ
$A_{AMQ}=\frac{AM*AQ}{2}=\frac{\frac{l}{2}*\frac{5l}{6}}{2}=\frac{5l^{2}}{24}$
$A_{BMN}=\frac{BM*BN}{2}=\frac{\frac{l}{2}*\frac{l}{3}}{2}=\frac{l^{2}}{12}$
$A_{CNP}=\frac{CN*CP}{2}=\frac{\frac{2l}{3}*\frac{l}{4}}{2}=\frac{l^{2}}{12}$
$A_{DQP}=\frac{DQ*DP}{2}=\frac{\frac{l}{6}\frac{3l}{4}}{2}=\frac{l^{2}}{16}$
Dupa cum ne uitam in figura de mai jos aria lui MNPQ este aria patratului minus ariile triunghiurilor calculate de mai sus
$A_{MNPQ}=A_{ABCD}-A_{AMQ}-A_{BMN}-A_{CNP}-A_{DQP}=l^{2}-\frac{5l^{2}}{24}-\frac{l^{2}}{12}-\frac{l^{2}}{12}-\frac{l^{2}}{16}=\frac{48l^{2}-2*5l^{2}-4*l^{2}-4*l^{2}-3*l^{2}}{48}=\frac{27*l*l}{48}=\frac{27*24*24}{48}=27*12=324$