Question

In triunghiul dreptunghic ABC, m(∡A) = 90, se cunosc AB = 8 cm, AC = 6 cm si BC = 10 cm. [AM] este mediana, M ∈ (BC), iar (BD este bisectoarea unghiului B, D e (AC). Daca AM ∩ BD = {N}, calculati AD/DC si AN/AM.

Answer 1

$a)Aplicam \ teorema \ bisectoarei \ in \ \Delta ABC:\frac{AD}{DC}=\frac{BA}{BC}=\frac{8}{10}=\frac{4}{5}$

$b)AM \ mediana=\textgreater BM=MC=\frac{BC}{2}=5cm$

$AM \ mediana=\textgreater AM=\frac{BC}{2}=5cm$

$AM=AN+NM=\textgreater NM=AM-AN$

$Aplicam \ teorema \ bisectoarei \ in \ \Delta BAM:\frac{AN}{NM}=\frac{BA}{BM}$

$\frac{AN}{AM-AN}=\frac{BA}{BM}$

$BA(AM-AN)=AN*BM$

$BA*AM-BA*AN=AN*BM$

$BA*AM=AN*BM+BA*AN$

$BA*AM=AN(BM+BA)$

$AN=\frac{BA*AM}{BM+BA}=\frac{40}{13}cm$

$\frac{AN}{AM}=\frac{\frac{40}{13}}{5}=\frac{40}{65}=\frac{8}{13}$