Question

A (-1,-1)
B (-1,1)
C (2,3)
D (2,-1)

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

In the image, polgon ABCD is dilated to form polygon A' B' C' D' with respect to the origin. What is the magnitude of dilation?

a. 1
b. 3
c. 2
d. 1/2

Answer 1

To find the magnitude of dilation, we can use the formula for dilation about the origin. The formula states that the coordinates of a point \( P(x, y) \) after dilation by a factor of \( k \) will be \( P'(kx, ky) \).

We can find the dilation factor \( k \) by comparing the coordinates of corresponding points in the original polygon and the dilated polygon. Let's take one pair of corresponding points. Using points \( A \) and \( A' \):
- Original point \( A \): \((-1, -1)\)
- Dilated point \( A' \): \((-2, -2)\)

Now we calculate the dilation factor \( k \):

\[
A'\ = k \cdot A
\]
\[
(-2, -2) = k \cdot (-1, -1)
\]

This gives us the equations:
- \( -2 = k \cdot (-1) \)
- \( -2 = k \cdot (-1) \)

From either equation, we can solve for \( k \):

\[
k = \frac{-2}{-1} = 2
\]

Now we can confirm this dilation factor with another pair of corresponding points, say points \( B \) and \( B' \):
- Original point \( B \): \((-1, 1)\)
- Dilated point \( B' \): \((-2, 2)\)

Calculating \( k \) again:

\[
(-2, 2) = k \cdot (-1, 1)
\]

This gives us:
- \( -2 = k \cdot (-1) \) leads to \( k = 2 \)
- \( 2 = k \cdot 1 \) also leads to \( k = 2 \)

Since the dilation factor is consistent across multiple points, we conclude that the magnitude of dilation is

**c. 2**.