Question

7 triangle fgh is transformed in the coordinate plane to form the image fgh. label the vertices of fgh and tell what single transformation was used.

Explanation:

Step1: Observe the transformation

By comparing the positions of $\triangle FGH$ and $\triangle F'G'H'$ in the coordinate - plane, we can see that the orientation of the triangle has changed. The transformation is a rotation.

Step2: Determine the center and angle of rotation

The center of rotation is the origin $(0,0)$. By observing the vertices, we can see that each point of $\triangle FGH$ is rotated $180^{\circ}$ counter - clockwise about the origin to get the corresponding point of $\triangle F'G'H'$.
If the coordinates of a point $(x,y)$ are rotated $180^{\circ}$ counter - clockwise about the origin, the new coordinates $(x',y')$ are given by $(x',y')=(-x,-y)$.
Let's assume the coordinates of $F$ are $(x_F,y_F)$, $G$ are $(x_G,y_G)$ and $H$ are $(x_H,y_H)$. Then the coordinates of $F'$ are $(-x_F,-y_F)$, $G'$ are $(-x_G,-y_G)$ and $H'$ are $(-x_H,-y_H)$.

Answer:

The transformation is a $180^{\circ}$ counter - clockwise rotation about the origin. To label the vertices of $F'G'H'$, if $F(x_1,y_1)$, then $F'(-x_1,-y_1)$; if $G(x_2,y_2)$, then $G'(-x_2,-y_2)$; if $H(x_3,y_3)$, then $H'(-x_3,-y_3)$. (Since the original coordinates of $F$, $G$ and $H$ are not given numerically, we express the vertices of the image in terms of the original vertices' coordinates after the $180^{\circ}$ rotation rule).