Question

no triângulo ABC o segmento DE é paralelo ao lado BC e AF é bissetriz relativa ao vértice A.determine as medidas dos segmentos BC e BD​

Answer 1

$\mathsf{\triangle\,ABC\sim\,\triangle\,ADE}$

$\dfrac{\overline{AB}}{\overline{AD}}=\dfrac{\overline{AC}}{\overline{AE} }$

$\dfrac{6+\overline{BD}}{6}=\dfrac{12}{9}\\54++9\overline{BD}=72$

$9\overline{BD}=72-54\\9\overline{BD}=18\\\overline{BD}=\dfrac{18}{9}=2$

Pelo teorema da bissetriz interna temos

$\dfrac{\overline{AB}}{\overline{AC}}=\dfrac{\overline{BF}}{BC}$

$\dfrac{8}{12}=\dfrac{\overline{BF}}{\overline{BF}+6}\\12\overline{BF}=8\overline{BF}+48\\12\overline{BF}-8\overline{BF}=48$

$4\overline{BF}=48\\\overline{BF}=\dfrac{48}{4}=12$

$\boxed{\boxed{\mathsf{\overline{BC}=12+6=18}}}$