Question

2
7
C' (4,6)
5
4
C (2,3)
B'(-2,2)
3
2
B (-1,1)
O
x
-5 -4 -3 -2 -1 0
3 4 5 6 7
A (-1 ,- 1)
=2
D'(4 ,- 2)
A'(-2 ,- 2)
D (2 ,- 1)
-3
In the image, polygon ABCD
is dilated to form polygon A′B′C′D′
with respect to the origin. What is the magnitude of dilation?

(1 point)
Responses

2
2

12
Start Fraction 1 over 2 End Fraction

1
1

3

Answer 1

To determine the magnitude of the dilation from polygon ABCD to polygon A'B'C'D', we can compare the coordinates of the corresponding points in both polygons.

Given the coordinates:

- A (-1, -1) to A' (-2, -2)
- B (-1, 1) to B' (-2, 2)
- C (2, 3) to C' (4, 6)
- D (2, -1) to D' (4, -2)

We can find the dilation factor \( k \) using the following formula for corresponding points:

\[
k = \frac{\text{distance from origin to point } P'}{\text{distance from origin to point } P}
\]

Calculating each:

1. For A and A':
- Distance from origin to A = √((-1)^2 + (-1)^2) = √2
- Distance from origin to A' = √((-2)^2 + (-2)^2) = √8 = 2√2
- Dilation factor (k_A) = \(\frac{2\sqrt{2}}{\sqrt{2}} = 2\)

2. For B and B':
- Distance from origin to B = √((-1)^2 + (1)^2) = √2
- Distance from origin to B' = √((-2)^2 + (2)^2) = √8 = 2√2
- Dilation factor (k_B) = \(\frac{2\sqrt{2}}{\sqrt{2}} = 2\)

3. For C and C':
- Distance from origin to C = √((2)^2 + (3)^2) = √13
- Distance from origin to C' = √((4)^2 + (6)^2) = √52 = 2√13
- Dilation factor (k_C) = \(\frac{2\sqrt{13}}{\sqrt{13}} = 2\)

4. For D and D':
- Distance from origin to D = √((2)^2 + (-1)^2) = √5
- Distance from origin to D' = √((4)^2 + (-2)^2) = √20 = 2√5
- Dilation factor (k_D) = \(\frac{2\sqrt{5}}{\sqrt{5}} = 2\)

In all cases, the dilation factor is consistently 2. Therefore, the magnitude of dilation is:

**2**