Question

Cho A(-2;3), A'(1;5), B(5;-3), B'(7;-2). Phép quay tâm I(x;y) biến A thành A' và B thành B'. Ta có x+y=?

Answer 1

Phép quay tâm I(x;y) biến A thành A' và B thành B'nên ta có:

\(\left\{{}\begin{matrix}IA=IA'\\IB=IB'\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(x+2\right)^2+\left(y-3\right)^2=\left(x-1\right)^2+\left(y-5\right)^2\\\left(x-5\right)^2+\left(y+3\right)^2=\left(x-7\right)^2+\left(y+2\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2+4x-6y+13=x^2+y^2-2x-10y+26\\x^2+y^2-10x+6y+34=x^2+y^2-14x+4y+53\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}6x+4y-13=0\\4x+2y-19=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\frac{25}{2}\\y=-\frac{31}{2}\end{matrix}\right.\Rightarrow x+y=-3\)