Question

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)

Answer 1

To find the scale factor needed to transform rectangle ABCD into rectangle EFGH using the given corner points, we first need to calculate the side lengths of both rectangles.

**Rectangle ABCD:**
- Points:
- A(-10, 6)
- B(-10, -2)
- C(-6, -2)
- D(-6, 6)

1. **Calculate the lengths of the sides of rectangle ABCD:**
- Length AB (vertical side) = |y-coordinate A - y-coordinate B| = |6 - (-2)| = |6 + 2| = 8
- Length AD (horizontal side) = |x-coordinate A - x-coordinate D| = |-10 - (-6)| = |-10 + 6| = 4

2. **Side lengths of rectangle ABCD:**
- Length = 8 (vertical), Width = 4 (horizontal)

**Rectangle EFGH:**
- Points:
- E(-5, 3)
- F(-5, -1)
- G(-3, -1)
- H(-3, 3)

1. **Calculate the lengths of the sides of rectangle EFGH:**
- Length EF (vertical side) = |y-coordinate E - y-coordinate F| = |3 - (-1)| = |3 + 1| = 4
- Length EH (horizontal side) = |x-coordinate E - x-coordinate H| = |-5 - (-3)| = |-5 + 3| = 2

2. **Side lengths of rectangle EFGH:**
- Length = 4 (vertical), Width = 2 (horizontal)

**Finding the Scale Factor:**
The scale factor can be determined by comparing the corresponding side lengths of the two rectangles:

- For the vertical sides:
\[
\text{Scale factor} = \frac{\text{Length of EFGH}}{\text{Length of ABCD}} = \frac{4}{8} = \frac{1}{2}
\]

- For the horizontal sides:
\[
\text{Scale factor} = \frac{\text{Width of EFGH}}{\text{Width of ABCD}} = \frac{2}{4} = \frac{1}{2}
\]

Since both calculations yield the same scale factor, the transformation scale factor from rectangle ABCD to rectangle EFGH is **\(\frac{1}{2}\)**.