Question

Trapezul ABCD are AB || CD , AB = 15 cm, CD = 10 cm, AD = 6 cm si BC = 8 cm . DacaAD n BC = {E} , calculati perimetrul EDC

Answer 1

Răspuns:

Perimetrul $P_{\Delta EDC}$ = 38 cm.

Explicație pas cu pas:

Datele problemei:

ABCD trapez, AB || CD;

AB = 15 cm, CD = 10 cm, AD = 6 cm, BC = 8 cm;

AD ∩ BC = {E};

Obs: Am atasat si o poza cu desenul.

Rezolvare:

$\left. \begin{array}{ll} ABCD \ trapez, \ AB || CD\\ AD \ \cap \ BC = {E} \end{array}\right \} = > \Delta DEC \sim \Delta AEB \\\\\\ Putem \ construi \ urmatorul \ sir \ de \ rapoarte: \\ \\\frac{ED}{EA} = \frac{EC}{EB} = \frac{DC}{AB} \\\\\frac{ED}{EA} = \frac{EC}{EB} = \frac{10}{15} = \frac{2}{3} \\\\Dar: \\EA = ED + DA = ED + 6\\EB = EC + CB = EC + 8\\Inlocuim\ in\ egalitatea\ de\ rapoarte\ si\ obtinem:$

$\frac{ED}{ED + 6} = \frac{EC}{EC + 8} = \frac{2}{3} = > \\\\\\\frac{ED}{ED + 6} = \frac{2}{3} \ \ \ \ si \ \ \frac{EC}{EC + 8} = \frac{2}{3} \\\\\\\frac{ED}{ED + 6} = \frac{2}{3} = > 3ED = 2(ED + 6) = > 3ED = 2ED + 12 = > ED = 12 cm\\\\\frac{EC}{EC + 8} = \frac{2}{3} = > 3EC = 2(EC + 8) = > 3EC = 2EC + 16 = > EC = 16cm\\\\Obs: Perimetrul\ unei\ figuri\ geometrice\ reprezinta\ suma\ lungimilor\ laturilor.\\\\= > P_{\Delta EDC} = ED + DC + EC = 12 + 10 + 16 = 38 cm$

Succes!