Question

Cho a + b + c = a^2 + b^2 + c^2 = 2 và x : y : z = a : b : c.

Chứng minh rằng: (x + y + z)^2 = 2x^2 + 2y^2 + 2z^2.
Cần gấp

m ọi người ơi :))))))

 

Answer 1

-Có: \(x:y:z=a:b:c\Rightarrow\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}=\dfrac{x+y+z}{a+b+c}=\dfrac{x+y+z}{2}\)

\(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\Rightarrow\dfrac{x^2}{a^2}=\dfrac{y^2}{b^2}=\dfrac{z^2}{c^2}=\dfrac{x^2+y^2+z^2}{a^2+b^2+c^2}=\dfrac{x^2+y^2+z^2}{2}\)

\(\Rightarrow\dfrac{\left(x+y+z\right)^2}{4}=\dfrac{x^2+y^2+z^2}{2}\)

\(\Rightarrow\left(x+y+z\right)^2=2x^2+2y^2+2z^2\left(đpcm\right)\)

Answer 2

Ta có x:a=y:b=z:c=x+y+z:a+b+c=x+y+z( vì a+b+c=1)do đó (x+y+z)^2=x^2:a^2=y^2:b^2=z^2:c^2=x^2+y^2+z^2:a^2+b^2+ c^2=x^2+y^2+z^2( vì a^2+b^2+c^2)Vậy (x+y+z)^2=x^2+y^2+z^2