Question

The triangle G'H'I' is a dilation of the triangle GHI. What is the scale factor of the dilation?
G' (-2,-2)
H' (-2, -1)
I' (0,-1)
Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Answer 1

To find the scale factor of the dilation from triangle GHI to triangle G'H'I', we need the coordinates of the original triangle GHI as well as those of the dilated triangle G'H'I'.

Given the coordinates for the dilated triangle:
- G' = (-2, -2)
- H' = (-2, -1)
- I' = (0, -1)

Let's assume the coordinates of triangle GHI are G = (x_G, y_G), H = (x_H, y_H), and I = (x_I, y_I).

The scale factor \(k\) can be calculated by taking the ratio of the distance from the center of dilation to the vertices of the original triangle GHI to the distance from the center of dilation to the vertices of the dilated triangle G'H'I'.

However, we need more details about the original triangle GHI to compute the scale factor.

If we assume that the center of dilation is at the origin (0,0), we can calculate the scale factor using a vertex of GHI.

Let's calculate the distance from the origin to one of the vertices of the dilated triangle and equate it to the distance from the origin to a corresponding vertex of the original triangle.

Let's assume the coordinates of GHI are:

- \( G = (g_x, g_y) \)
- \( H = (h_x, h_y) \)
- \( I = (i_x, i_y) \)

To calculate the scale factor:

1. Calculate the distance from the origin to the dilated triangle:
\[
\text{Distance to } G' = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
\[
\text{Distance to } H' = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
\[
\text{Distance to } I' = \sqrt{(0)^2 + (-1)^2} = \sqrt{1} = 1
\]

2. Now, select corresponding vertices of GHI (not provided). Let's say the original triangle had coordinates \( G = (-1, -1), H = (-1, 0), I = (0, 0) \) (for calculation purposes).

Therefore, using the points we assume:
- Distance to G = \(\sqrt{(-1)^2 + (-1)^2} = \sqrt{2}\)
- Distance to H = \(\sqrt{(-1)^2 + (0)^2} = 1\)
- Distance to I = \(\sqrt{(0)^2 + (0)^2} = 0\)

Now, we use corresponding pairs to calculate \(k\):

1. For G and G':
\[
k = \frac{D(G')} {D(G)} = \frac{2\sqrt{2}}{\sqrt{2}} = 2
\]

2. For H and H':
\[
k = \frac{\sqrt{5}}{1} = \sqrt{5}
\]

3. For I and I':
\[
k = \frac{1}{0} (undefined)
\]

If the original triangle had coordinates where the vertices can be established or if I made the wrong calculations with the scale \(k = \frac{\text{Distance to G'}}{\text{Distance to G}}\) it can yield the scale factor, which resolves to a unified value. The above calculations relied on arbitrary assumptions for GHI.

If you can provide actual coordinates of GHI, we can calculate the exact scale factor based on those as well. Given a scenario with GHI, check for distance ratios; each should yield the same scale factor \(k\).

Exploring proper fractions and simplifications, \(k\) would either remain as a numeric fraction or integer based on those inputs.

Please share triangle GHI or clarify further, and I can assist with determining the precise scale factor.

In practice, if presuming original vertex triangles as starting standard forms let's assume a simple triangle for calculation \(k = 1\) is common in simplicity (1:1 transformation).

So the scale factor would be represented here as:

\[
\text{Hypothetical Scale Factor} = \frac{r_1}{r_2} = \frac{Distance G'}{Distance G} = \frac{\text{Point φ' G'}}{\text{Point φ G}}.
\]

Could be resolved into standard fashion and depend on accurate historical points basis to assure dilation function.

Now, please provide GHI vertices for definitive work!