- Apply the dilation formula \((x', y') = (Kx, Ky)\) to the given points \((2, 4)\) and \((8, 16)\).
- Solve the equation \(8 = 2K\) for \(K\) to find \(K = 4\).
- Compare \(K=4\) with \(1\) to determine the type of transformation.
- Conclude that since \(K > 1\), the dilation is an expansion, with \(\boxed{K=4}\).
Explanation
- Define the transformation rule and identify given points.
A dilation centered at the origin \(O=(0,0)\) with a scale factor \(K\) maps any point \((x, y)\) to a new point \((x', y')\) according to the transformation rule:
\[x' = Kx\]
\[y' = Ky\]
Given that the point \(P=(2, 4)\) is mapped to \(P'=(8, 16)\), we can substitute these coordinates into the transformation equations to determine the value of \(K\).
- Calculate the scale factor \(K\).
Using the x-coordinates:
\[8 = K \cdot 2\]
Solving for \(K\):
\[K = \frac{8}{2} = 4\]
Using the y-coordinates:
\[16 = K \cdot 4\]
Solving for \(K\):
\[K = \frac{16}{4} = 4\]
Both coordinates yield the same scale factor, confirming that \(K=4\).
- Determine the type of dilation.
A dilation is classified based on the absolute value of the scale factor \(K\):
- If \(|K| > 1\), the transformation is an expansion.
- If \(0 < |K| < 1\), the transformation is a contraction.
Since we found \(K = 4\), and \(4 > 1\), the dilation is an expansion. The image point \(P'\) is four times further from the origin than the original point \(P\). The final value for the scale factor is \(K = 4\) and it is an expansion.
Examples
Imagine you are using a projector to display an image on a wall. If the distance of the projected image from the light source is 4 times larger than the original slide's dimensions, you are applying a dilation with a scale factor of 4. This is an expansion, making the image much larger for the audience to see! Calculating the scale factor allows you to adjust the projector settings to fit the image perfectly within the screen's frame.