Question

5.2 dilation or violation?
heres a triangle abc. points m and n are the midpoints of 2 sides.

1. convince yourself triangle abc is a dilation of triangle amn.

what is the center of dilation?
point
what is the scale factor?

Explanation:

Step1: Recall the concept of dilation center

In a dilation, the center of dilation is the fixed point. For triangle \(ABC\) and its dilation \(AMN\), by observing the figure (since \(M\) and \(N\) are mid - points and the lines connecting corresponding vertices of \(\triangle ABC\) and \(\triangle AMN\) (i.e., \(A - A\), \(B - M\), \(C - N\)) intersect at point \(A\)), the center of dilation is point \(A\).

Step2: Recall the mid - point theorem and scale factor formula

Since \(M\) is the mid - point of \(AB\) and \(N\) is the mid - point of \(AC\), by the mid - point theorem, \(MN=\frac{1}{2}BC\), \(AM = \frac{1}{2}AB\), and \(AN=\frac{1}{2}AC\). The scale factor \(k\) of a dilation is given by the ratio of the lengths of corresponding sides of the dilated figure to the original figure. Here, if we consider \(\triangle ABC\) as the dilated figure and \(\triangle AMN\) as the original figure, \(k = 2\) (because \(AB = 2AM\), \(AC=2AN\), \(BC = 2MN\)).

Answer:

Center of dilation: Point \(A\)
Scale factor: \(2\)