Question

Exercitiile incercuite din puncte

Answer 1

1. Aflam coordonatele centrului de greutate

$G(\frac{x_A+x_B+x_C}{3},\frac{y_A+y_B+y_C}{3} )\\\\G(\frac{8}{3} ,5)$

O(0,0)

$d(O,G)=\sqrt{(x_G-x_O)^2+(y_G-y_O)^2} =\sqrt{\frac{64}{9} +25} =\sqrt{\frac{289}{9} } =\frac{17}{3}$

$OM=\frac{1}{2} \times \frac{17}{3} =\frac{17}{6}$

2. a,b,c progresie aritmetica

r=1

$OA=\sqrt{(x_A-x_O)^2+(y_A-y_O)^2} =\sqrt{a^2+(a+1)^2} \\\\OB=\sqrt{(x_B-x_O)^2+(y_B-y_O)^2} =\sqrt{b^2+(b+1)^2}\\\\OC=\sqrt{(x_C-x_O)^2+(y_C-y_O)^2} =\sqrt{c^2+(c+1)^2}$

• Stim ca OA²+OC²-9=OB²

a²+(a+1)²+c²+(c+1)²-9=b²+(b+1)²

• Mai stim ca a,b,c, sunt in progresie cu r=1

a=a

b=a+1

c=b+1=a+2

• Vom inlocui si vom obtine:

a²+(a+1)²+c²+(c+1)²-9=(a+1)²+c²

Se reduce (a+1)² si c²

a²+(a+3)²-9=0

a²+a²+6a+9-9=0

2a²+6a=0

2a(a+3)=0

a=0⇒ b=1 si c=2

a=-3⇒ b=-2⇒ c=-1