Question

move one answer choice into each box to make the statement true.
a dilation centered at point maps point c to point a.

Explanation:

Step1: Recall the concept of dilation

Dilation is a transformation that enlarges or reduces a figure. The formula for dilation of a point \(P(x,y)\) centered at \(O(a,b)\) with scale factor \(k\) is \(P'(x',y')=(a + k(x - a),b + k(y - b))\). In the context of points on a line (assuming a one - dimensional or coordinate - like relationship for simplicity), if we consider the distance from the center of dilation.
Let's assume the distance from the center of dilation to \(C\) and then to \(A\). If we consider the ratio of the distances.
Suppose we check the center. If the center is \(E\). Let's assume the "distance" (in a geometric sense, could be along a line segment) from \(E\) to \(C\) and \(E\) to \(A\).
Let the scale factor \(k=\frac{EA}{EC}\). If we assume a simple geometric relationship (by visual inspection or given the nature of the problem setup), if we consider the relative positions of the points.
If we assume that the center of dilation is \(E\). Let's say \(EC\) is twice \(EA\) (in the sense of the length relationship for dilation). The scale factor \(k = \frac{1}{2}\) because \(A\) is closer to \(E\) than \(C\) is. Mathematically, if we think of the vector or length from the center \(E\) to \(C\) and \(E\) to \(A\), \(EA=\frac{1}{2}EC\) (assuming a linear relationship between the points with respect to \(E\)).

Answer:

A dilation centered at point \(E\) with scale factor \(\frac{1}{2}\) maps point \(C\) to point \(A\). So the first box is \(E\) and the second box is \(\frac{1}{2}\)