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

Question

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:

GPT-4o mini · Answer

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: $Length = | y_{A} - y_{D} | = | 6 - 6 | = 0 (this is incorrect as A and D lie on the same side, we mean | - 10 - (- 6) |)$ Since AD is horizontal (points A and D have same y-coordinate): $Length = | - 10 - (- 6) | = | - 10 + 6 | = 4$
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): $Width = | y_{A} - y_{B} | = | 6 - (- 2) | = | 6 + 2 | = 8$

So, the dimensions of rectangle ABCD are:

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): $Length = | - 5 - (- 3) | = | - 5 + 3 | = 2$
The width can be measured between E and F (vertical): $Width = | y_{E} - y_{F} | = | 3 - (- 1) | = | 3 + 1 | = 4$

So, the dimensions of rectangle EFGH are:

Ratio Calculation:

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

The scale factor for length: ${Scale Factor}_{l e n g t h} = \frac{Length of ABCD}{Length of EFGH} = \frac{4}{2} = 2$
The scale factor for width: ${Scale Factor}_{w i d t h} = \frac{Width of ABCD}{Width of EFGH} = \frac{8}{4} = 2$

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

$Scale Factor = 2$

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