To determine the magnitude of dilation applied to triangle ABC to result in triangle A'B'C', we can use the coordinates of a corresponding point from triangle ABC and triangle A'B'C'.
The formula for dilation is given by:
\[ A' = k \cdot A \]
where \(k\) is the scale factor (dilation magnitude), \(A\) is the original point, and \(A'\) is the dilated point. We can use the coordinates of point A and A' for our calculations:
- Calculate the dilation magnitude using A and A':
- Original A: \((4, 2)\)
- Dilated A': \((2, 1)\)
We can set up the following equations using the dilation formula:
\[ x' = k \cdot x \] \[ y' = k \cdot y \]
From A to A':
\[ 2 = k \cdot 4 \] \[ 1 = k \cdot 2 \]
Now, we can solve for \(k\) in each equation:
-
From the first equation: \[ k = \frac{2}{4} = \frac{1}{2} \]
-
From the second equation: \[ k = \frac{1}{2} = \frac{1}{2} \]
Both calculations give the same result, so we can conclude that the dilation factor \(k\) is \(\frac{1}{2}\).
To further ensure accuracy, we can verify with points B and B', and C and C':
For B and B':
- B: \((-2, 6)\)
- B': \((-1, 3)\)
\[ -1 = k \cdot (-2) \quad \Rightarrow \quad k = \frac{1}{2} \] \[ 3 = k \cdot 6 \quad \Rightarrow \quad k = \frac{1}{2} \]
For C and C':
- C: \((-6, -4)\)
- C': \((-3, -2)\)
\[ -3 = k \cdot (-6) \quad \Rightarrow \quad k = \frac{1}{2} \] \[ -2 = k \cdot (-4) \quad \Rightarrow \quad k = \frac{1}{2} \]
Since all points confirm that the dilation magnitude is \(\frac{1}{2}\), the answer is:
\[ \text{Dilation magnitude} = \frac{1}{2} \]
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