Question

there is a triangle abc. m is \\( \frac { 1 } { 3 } \\) of the way from a to b. n is \\( \frac { 1 } { 3 } \\) of the way from a to c. 1. convince yourself triangle amn is a dilation of triangle abc what is the center of dilation? point what is the scale factor?

Explanation:

Step1: Determine the center of dilation

In a dilation, the center of dilation is the fixed point from which all points are expanded or contracted. For triangle \(AMN\) being a dilation of triangle \(ABC\), since \(A\) is a common vertex and the lines \(AM\) and \(AN\) are parts of the transformation lines from \(A\) to \(B\) and \(A\) to \(C\) respectively, the center of dilation is point \(A\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the lengths of corresponding sides. \(M\) is \(\frac{1}{3}\) of the way from \(A\) to \(B\), so \(AM=\frac{1}{3}AB\). Similarly, for the corresponding sides in the dilation of \(\triangle AMN\) and \(\triangle ABC\), if we consider the side \(AM\) and \(AB\) (or \(AN\) and \(AC\) since the ratio will be the same for a dilation), the scale factor \(k=\frac{AM}{AB}\). Since \(AM = \frac{1}{3}AB\), \(k=\frac{1}{3}\)

Answer:

Center of dilation: \(A\); Scale factor: \(\frac{1}{3}\)