Question

1. part a: describe the transformation that maps def to def. part b: algebraically represent the transformation needed to map def to def.

Explanation:

Part A:

Step1: Analyze the transformation type

By observing the graph, we can see that the size of the triangle DEF is reduced to get triangle D'E'F'. This is a dilation transformation.

Part B:

Step1: Find the scale factor

Let's assume a point. For example, if we consider the horizontal distance from \(D\) to \(F\) and from \(D'\) to \(F'\). Suppose the coordinates of \(D=(2,1)\), \(F=(5,6)\), \(D'=(4,1)\), \(F'=(5,4)\). The length of \(DF\) using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(d_{DF}=\sqrt{(5 - 2)^2+(6 - 1)^2}=\sqrt{9 + 25}=\sqrt{34}\). The length of \(D'F'\) is \(d_{D'F'}=\sqrt{(5 - 4)^2+(4 - 1)^2}=\sqrt{1+9}=\sqrt{10}\) (another way: if we consider the ratio of corresponding side lengths. Looking at the horizontal segments: from \(D\) to \(E\) (assuming \(E\) is a point on the base) and \(D'\) to \(E'\). The base of \(\triangle DEF\) has a length of \(4\) units (say from \(x = 2\) to \(x=6\)) and the base of \(\triangle D'E'F'\) has a length of \(2\) units (from \(x = 4\) to \(x = 6\)). The scale factor \(k=\frac{1}{2}\). The algebraic representation of a dilation centered at the origin is \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)

Answer:

Part A:

The transformation is a dilation (a reduction).

Part B:

\((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)