Question

triunghiul ABC
BM/AM=3/5
N apartine de Ac
NC=12cm
AC=32cm

Calc: A Amn/A ABC ,stiind ca Bc=48cm



A-aria

Answer 1

$\frac{BM}{AM} = \frac{3}{5}$ si

AN=AC-NC=32-12=20 cm, deci:

$\frac{CN}{AN} = \frac{12}{20} = \frac{3}{5}$

Observam ca $\frac{BM}{AM} = \frac{CN}{AN} = \frac{3}{5}$, deci MN determina pe AB si AC segmente proportionale, deci MN || BC si deci ΔAMN≈ΔABC, iar raportul ariilor celor doua triunghiuri este egal cu patratul raportului de asemanare, adica:

$\frac{Aria AMN}{Aria ABC} = ( \frac{AN}{AC} )^{2}$

$\frac{Aria AMN}{Aria ABC} = ( \frac{20}{32} )^{2}$

$\frac{Aria AMN}{Aria ABC} = ( \frac{5}{8} )^{2}$

$\frac{Aria AMN}{Aria ABC} = \frac{25}{64}$