Question

in this diagram, point d is rotated 180° using b as the center. explain why the image of d must lie on the ray ba.

Explanation:

Step1: Recall rotation property

A 180 - degree rotation about a point $B$ maps a point $D$ to a point $D'$ such that $B$ is the mid - point of the line segment $\overline{DD'}$.

Step2: Analyze collinearity

Points $B$, $D$ and the image of $D$ (say $D'$) are collinear with $B$ in the middle. Since the ray $BA$ is on the line passing through $B$ and $D$, when rotated 180 degrees about $B$, the image of $D$ will lie on the opposite side of $B$ on the same line, which is the ray $BA$.

Answer:

A 180 - degree rotation about a point $B$ results in the pre - image and image being on a line with $B$ as the mid - point. Since $D$ lies on the line containing $B$ and $A$, its 180 - degree rotation about $B$ places its image on the ray $BA$.