Question

what dilation is it when A(-4, 3) to E(4.5, 5), B(-3, 1) to F(1.6, 3.8)

Answer 1

what translation and dilation is it when A(-4, 3) to E(4.5, 5), B(-3, 1) to F(1.6, 3.8)

Answer 2

To determine the dilation that maps points A to E and B to F, we'll first need to calculate the scale factor of the dilation.

The dilation can be represented by a scale factor \( k \) and a center of dilation \( (x_0, y_0) \). The formula for dilation is:

\[
(x', y') = (x_0 + k(x - x_0), y_0 + k(y - y_0))
\]

Where:
- \( (x, y) \) are the original coordinates,
- \( (x', y') \) are the new coordinates after dilation,
- \( k \) is the scale factor,
- \( (x_0, y_0) \) is the center of dilation.

Given the points A(-4, 3) to E(4.5, 5) and B(-3, 1) to F(1.6, 3.8), we can use these points to find \( k \) and the possible center of dilation.

### Step 1: Calculate the scale factor \( k \) using points A to E.

1. For A(-4, 3) to E(4.5, 5):
- We can express the coordinates as:
- \( x' = k(x + 4) - 4 \) (x-coordinate)
- \( y' = k(y - 3) + 3 \) (y-coordinate)

Rearranging gives us:
- \( x' = x + 4k - 4 \)
- \( y' = y - 3 + 3k \)

2. Setting up the equations with points A and E:
- For x-coordinate:
\( 4.5 = -4 + k(4 + 4) \)
- \( 4.5 = -4 + 8k \)
- \( 8k = 8.5 \)
- \( k = 1.0625 \)

- For y-coordinate:
\( 5 = 3 + k(3 - 3) \)
- \( 5 = 3 + 0 \)
- This equation holds true regardless of \( k \) (no information here)

### Step 2: Validate with B to F.

1. For B(-3, 1) to F(1.6, 3.8):
- Setting up the equations similarly:
- For x-coordinate:
\( 1.6 = -3 + k(3 + 3) \)
- \( 1.6 = -3 + 6k \)
- \( 6k = 4.6 \)
- \( k \approx 0.7667 \)

- For y-coordinate:
\( 3.8 = 1 + k(1 - 1) \)
- \( 3.8 = 1 + 0 \)
- This equation also holds (again, no information here)

### Conclusion:

The calculations for the scaling suggest an inconsistency, which might arise from a misleading assumption about the center. To fully analyze the dilation, we'd need to consider a common center of dilation \( (x_0, y_0) \).

However, it appears there may be an error, as the scale factor should theoretically be consistent across all points. If the dilation factors differ, it's possible these points do not undergo a simple linear dilation from a single center.

To finalize, rechecking points, calculating pooling among points or changing geographic centers may yield a consistent dilation result, but based on pointed calculations above, we derived scale factors of approximately \( k \approx 1.0625 \) and \( k \approx 0.7667 \) respectively, indicating differing transformations suggestive of either differing centers or non-linear scaling.

This would demand further checks — possible details unstated. If points have been confirmed valid through given transformations keeping consistent \( k \), more defined examination prevails appropriate.

Answer 3

To find the translation and dilation that maps points \( A(-4, 3) \) to \( E(4.5, 5) \) and \( B(-3, 1) \) to \( F(1.6, 3.8) \), we can analyze the transformations step by step.

### Step 1: Determine the Translation

Translation is a movement of points in a specific direction and distance. We can find the translation vector by calculating the difference between the starting and ending coordinates of the points.

**Translation from A to E:**
- \( A(-4, 3) \) to \( E(4.5, 5) \):
- Change in x: \( 4.5 - (-4) = 4.5 + 4 = 8.5 \)
- Change in y: \( 5 - 3 = 2 \)

Thus, the translation vector \( T \) is:
\[
T(8.5, 2)
\]

**Translation from B to F:**
- \( B(-3, 1) \) to \( F(1.6, 3.8) \):
- Change in x: \( 1.6 - (-3) = 1.6 + 3 = 4.6 \)
- Change in y: \( 3.8 - 1 = 2.8 \)

Thus, the translation vector \( T' \) is:
\[
T'(4.6, 2.8)
\]

### Translational Analysis:
The translation vectors \( (8.5, 2) \) and \( (4.6, 2.8) \) are not the same, indicating these points undergo different translational movements. Thus, it is crucial to start by proceeding with finding the center of dilation.

### Step 2: Determine the Center of Dilation and the Dilation Factor

Dilation changes the size of a figure but keeps the shape similar. The center of dilation \( (x_0, y_0) \) leads us to find a scaling factor \( k \).

**Let’s assume point E is \( k \) times away from point A.**

The general formula for dilation from a point \( (x_0, y_0) \) is:
\[
(x', y') = (x_0 + k(x - x_0), y_0 + k(y - y_0))
\]

To find \( k \) and the center of dilation, we can set up the equations using points A and E:

Let’s denote the center of dilation as \( (x_0, y_0) \). For point A:
- Dilation for \( x \):
\[
4.5 = x_0 + k(-4 - x_0) \tag{1}
\]
- Dilation for \( y \):
\[
5 = y_0 + k(3 - y_0) \tag{2}
\]

Next, for point B:
- Dilation for \( x \):
\[
1.6 = x_0 + k(-3 - x_0) \tag{3}
\]
- Dilation for \( y \):
\[
3.8 = y_0 + k(1 - y_0) \tag{4}
\]

### Step 3: Solve the Equations

To solve for \( (x_0, y_0) \) and \( k \):

1. **From equations (1) and (3)**:

Subtract equation (3) from (1):
\[
4.5 - 1.6 = k(-4 - x_0 + 3 + x_0)
\]
\[
2.9 = k(-1)
\]
Thus,
\[
k = -2.9 \tag{5}
\]

2. Using equation (5) in (1) to solve for \(x_0\):
Substituting \( k \) back into (1):
\[
4.5 = x_0 + (-2.9)(-4 - x_0)
\]
\[
4.5 = x_0 + 11.6 + 2.9x_0
\]
Simplifying gives:
\[
4.5 = 3.9x_0 + 11.6
\]
\[
-7.1 = 3.9x_0
\]
\[
x_0 \approx -1.82 \tag{6}
\]

3. Using \( k \) in (2):
\[
5 = y_0 + (-2.9)(3 - y_0)
\]
\[
5 = y_0 - 8.7 + 2.9y_0
\]
Simplifying gives:
\[
5 + 8.7 = 3.9y_0
\]
\[
13.7 = 3.9y_0
\]
\[
y_0 \approx 3.51 \tag{7}
\]

### Summary

From our computation, we arrive at:
- Center of Dilation \( (x_0, y_0) \approx (-1.82, 3.51) \)
- Dilation Factor \( k \approx -2.9 \)

Thus, the overall transformation consists of both translation and dilation characterized previously with positions noticed.

The final transformations are:
1. **Translation** vector \( T \) obtained from difference calculations (assuming uniform across points),
2. **Dilation:** with center at found point \( (-1.82, 3.51) \) inducing a scaling of \( k \approx -2.9 \), implying an inversion and enlargement.

Rechecking computations consistency aligns in transformations as clear, or adjustments may be further confirmed depending on additional specified characteristics.