Question

part b: the archeologist then rotates the fragment 180° about the origin to see it from a different angle. what are the new coordinates of the vertices?
part c: next, the archeologist moves the bone fragment back to its original position. then she dilates the image of the bone fragment using a scale factor of 1.5 and a center of dilation at the origin. what are the new coordinates of the vertices?

Explanation:

Step1: Recall rotation rule for 180 - degree rotation

The rule for rotating a point $(x,y)$ 180 - degrees about the origin is $(x,y)\to(-x,-y)$. For each vertex of the fragment, apply this rule.

Step2: Recall dilation rule

The rule for dilating a point $(x,y)$ with a scale factor $k$ and center of dilation at the origin is $(x,y)\to(kx,ky)$. Here $k = 1.5$. Multiply the $x$ and $y$ - coordinates of each vertex in the original position by 1.5.

Since no original coordinates of vertices are given, assume a vertex has original coordinates $(x,y)$.
For part (b):
The new coordinates after 180 - degree rotation about the origin are $(-x,-y)$.
For part (c):
The new coordinates after dilation with a scale factor of 1.5 and center of dilation at the origin are $(1.5x,1.5y)$.

Answer:

(b) If a vertex has original coordinates $(x,y)$, new coordinates after 180 - degree rotation about the origin are $(-x,-y)$.
(c) If a vertex has original coordinates $(x,y)$, new coordinates after dilation with scale factor 1.5 and center at the origin are $(1.5x,1.5y)$.