Question

Using the side lengths of rectangle ABCD and rectangle EFGH, find the scale factor needed to transform from one shape to the other. (Click Here for Help Video)

A=(-9,6)
B=(-10,-2)
C=(-6,-2)
D=(-6,6)
E=(-5,3)
F=(-5,-1)
G=(-3,-1)
H=(-3,3)

(2 points)
The scale factor needed to convert from ABCD to EFGH is:

The scale factor needed to convert from EFGH to ABCD is:

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Answer 1

To find the scale factor needed to transform from rectangle ABCD to rectangle EFGH, we first need to determine the dimensions (length and width) of each rectangle.

### Rectangle ABCD
Vertices of rectangle ABCD:
- A=(-9,6)
- B=(-10,-2)
- C=(-6,-2)
- D=(-6,6)

To find the lengths of the sides, we calculate the distances between vertices:

1. Length \(AB\) (the distance between points A and B):
\[
L_{AB} = \sqrt{(-10 - (-9))^2 + (-2 - 6)^2} = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}
\]

2. Length \(AD\) (the distance between points A and D):
\[
L_{AD} = \sqrt{(-6 - (-9))^2 + (6 - 6)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3
\]

So the dimensions of rectangle ABCD are approximately:
- Width \(L_{AB} = \sqrt{65} \approx 8.06\)
- Height \(L_{AD} = 3\)

### Rectangle EFGH
Vertices of rectangle EFGH:
- E=(-5,3)
- F=(-5,-1)
- G=(-3,-1)
- H=(-3,3)

Similarly, find the lengths of the sides:

1. Length \(EF\) (the distance between points E and F):
\[
L_{EF} = \sqrt{(-5 - (-5))^2 + (-1 - 3)^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4
\]

2. Length \(EH\) (the distance between points E and H):
\[
L_{EH} = \sqrt{(-3 - (-5))^2 + (3 - 3)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2
\]

So the dimensions of rectangle EFGH are:
- Width \(L_{EF} = 4\)
- Height \(L_{EH} = 2\)

### Finding the Scale Factors
To find the scale factor from rectangle ABCD to rectangle EFGH, we divide the dimensions of EFGH by the dimensions of ABCD:
1. Scale factor for width:
\[
\text{Scale factor width} = \frac{4}{\sqrt{65}} \approx \frac{4}{8.06} \approx 0.496
\]

2. Scale factor for height:
\[
\text{Scale factor height} = \frac{2}{3} \approx 0.667
\]

Since the dimensions may not scale equally, you can use either scale factor depending on how you want to transform from one shape to another.

For a consistent scale factor, you could choose to average these two results or find a method that defines how you want to transform.

To determine the scale factor from EFGH to ABCD, you would take the reciprocal of the previous factors.

### Answers:
The scale factor needed to convert from ABCD to EFGH is: \( \approx 0.496 \) for width and \( \approx 0.667 \) for height (you can choose based on your transformation needs).

The scale factor needed to convert from EFGH to ABCD is (reciprocals):
- For width: \( \frac{\sqrt{65}}{4} \approx 2.016\)
- For height: \( \frac{3}{2} = 1.5\)

You can state the conversions as:
- From ABCD to EFGH using \( 0.496\) for common scaling.
- From EFGH to ABCD using \(2.016\) or \(1.5\) respectively.