Question

problem 3: rotate △xyz 180°. verify xy = xy. rotate about origin answer:

Explanation:

Step1: Recall rotation rule

The rule for rotating a point $(x,y)$ 180 - degrees about the origin is $(x,y)\to(-x,-y)$. Let the coordinates of point $X=(x_1,y_1)$ and $Y=(x_2,y_2)$ in $\triangle XYZ$. After a 180 - degree rotation about the origin, the coordinates of $X'=(-x_1,-y_1)$ and $Y'=(-x_2,-y_2)$.

Step2: Use distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The distance $XY=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The distance $X'Y'=\sqrt{((-x_2)-(-x_1))^2+((-y_2)-(-y_1))^2}=\sqrt{(-(x_2 - x_1))^2+(-(y_2 - y_1))^2}$. Since $(-a)^2=a^2$ for any real number $a$, we have $X'Y'=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=XY$.

Answer:

The lengths $XY$ and $X'Y'$ are equal because a 180 - degree rotation about the origin is a rigid - motion transformation which preserves distances. Mathematically, using the distance formula and the rotation rule about the origin, we can show that the distance between two points before and after a 180 - degree rotation about the origin is the same.