Question

3) Fie ABC un triunghi dreptunghic cu m(A)=90° Sa se arate ca
a) sin²B + sin²C = 1
b) sin B + cos B = \frac{AB+AC}{BC}
c) AD²=AB · AC · sinB · sinC , unde AD⊥ BC si D ∈ [BC]

Answer 1

a) sin B=AC/BC
sin C=AB/BC
sin²B= AC²/BC²
sin²C=AB²/BC²
sin²B + sin ² C= (AC²+AB²)/BC²=BC²/BC²=1

b)sin B+cos B= AC/BC+AB/BC= (AC+AB)/BC

c) AD²= AB *AC * AC/BC*AB/BC= (AB*AC/BC)²
AD=AB*AC/BC (teorema inaltimii)
AD*BC=AB*AC /:2
AD*BC:2= AB*AC:2
A ABC=A ABC (adevarat)