Question

problem 6
here is triangle abc on the coordinate plane.
here is your work from the previous screens.
triangle ghi is a dilation of triangle def.
identify the center of dilation.
determine the scale factor.

Explanation:

Step1: Find the center of dilation

The center of dilation is the point that remains fixed during the dilation. By observing the corresponding points of the two triangles \(DEF\) and \(GHI\), we can see that the lines connecting corresponding vertices (e.g., \(D\) to \(G\), \(E\) to \(H\), \(F\) to \(I\)) intersect at the point \((-1,1)\).

Step2: Calculate the scale factor

Let's use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Take two corresponding points, say \(D(10,- 5)\) and \(G(3,1)\)
The distance from \(D\) to the center \((-1,1)\):

$$
LATEXBLOCK0
$$

The distance from \(G\) to the center \((-1,1)\):

$$
LATEXBLOCK1
$$

Take another pair of corresponding points, say \(E(-10,-5)\) and \(H(-2,1)\)
The distance from \(E\) to the center \((-1,1)\):

$$
LATEXBLOCK2
$$

The distance from \(H\) to the center \((-1,1)\):

$$
LATEXBLOCK3
$$

Alternatively, we can use the ratio of side - lengths.
If we consider the horizontal or vertical segments.
Let's assume we count the number of units.
For example, if we consider a horizontal segment. Suppose in \(\triangle DEF\) a horizontal segment has length \(12\) (from \(x=-10\) to \(x = 2\) in some base - like segment, assume a simple case of similar triangles with lattice - point based side - length counting). In \(\triangle GHI\) the corresponding horizontal segment has length \(4\).
The scale factor \(k=\frac{1}{3}\)

Answer:

Center of dilation: \((-1,1)\)
Scale factor: \(\frac{1}{3}\)