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Using the side lengths of rectangle ABCD and rectangle EFGH, find the scale factor needed to transform from one shape to the other

a=(-10,6)
b=(-10,-2)
c=(-6,-2)
d=(-6,6)
e=(-5,3)
f=(-5,-1)
g=(-3,-1)
h=(-3,3)
The scale factor needed to convert from EFGH to ABCD is:

Answers

Answered by GPT-4o mini
To find the scale factor needed to transform rectangle EFGH to rectangle ABCD, we first need to determine the side lengths of both rectangles.

**Finding the side lengths of rectangle ABCD:**

- The length of rectangle ABCD can be calculated using the coordinates of vertices A and D (or A and B).
- The distance between A and D:
\[
\text{Length} = |y_A - y_D| = |6 - 6| = 0 \quad (\text{this is incorrect as A and D lie on the same side, we mean} |-10 - (-6)|)
\]
Since AD is horizontal (points A and D have same y-coordinate):
\[
\text{Length} = |-10 - (-6)| = |-10 + 6| = 4
\]

- The width of rectangle ABCD can be calculated using the coordinates of vertices A and B or B and C.
- The distance between A and B (or B and C):
\[
\text{Width} = |y_A - y_B| = |6 - (-2)| = |6 + 2| = 8
\]

So, the dimensions of rectangle ABCD are:
- Length = 4
- Width = 8

**Finding the side lengths of rectangle EFGH:**

- The length of rectangle EFGH can similarly be calculated using points E and G and E and F.
- The distance between E and G (horizontal):
\[
\text{Length} = |-5 - (-3)| = |-5 + 3| = 2
\]

- The width can be measured between E and F (vertical):
\[
\text{Width} = |y_E - y_F| = |3 - (-1)| = |3 + 1| = 4
\]

So, the dimensions of rectangle EFGH are:
- Length = 2
- Width = 4

**Ratio Calculation:**

Now, we can find the scale factor from rectangle EFGH to rectangle ABCD:

- The scale factor for length:
\[
\text{Scale Factor}_{length} = \frac{\text{Length of ABCD}}{\text{Length of EFGH}} = \frac{4}{2} = 2
\]
- The scale factor for width:
\[
\text{Scale Factor}_{width} = \frac{\text{Width of ABCD}}{\text{Width of EFGH}} = \frac{8}{4} = 2
\]

Since both scale factors are consistent, the overall scale factor needed to convert from rectangle EFGH to rectangle ABCD is:

\[
\text{Scale Factor} = 2
\]

Thus, the scale factor needed to transform from EFGH to ABCD is **2**.
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