Bot GPT 4-o mini 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.
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