To find the scale factor of the dilation, you can compare the distances of corresponding vertices before and after dilation.
Let's calculate the distance of one vertex before and after dilation. We'll use the vertex \((4, 4)\) from the original quadrilateral and \((3, 3)\) from the dilated quadrilateral. The formula for the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Original Distance (from the origin \((0, 0)\) to \((4, 4)\)):
\[ d_{original} = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Dilated Distance (from the origin \((0, 0)\) to \((3, 3)\)):
\[ d_{dilated} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Calculate the Scale Factor:
To find the scale factor \(k\), divide the dilated distance by the original distance:
\[ k = \frac{d_{dilated}}{d_{original}} = \frac{3\sqrt{2}}{4\sqrt{2}} = \frac{3}{4} \]
Thus, the scale factor of the dilation is \(\frac{3}{4}\).