Example Question - dilation of a point
Here are examples of questions we've helped users solve.
Dilation of a Point
The question is asking for the image of the point (12, −4) after a dilation by a scale factor of \( \frac{1}{3} \), centered at the origin. To find the image of a point after dilation, you multiply the coordinates of the original point by the scale factor. Here the scale factor is \( \frac{1}{3} \). Thus, the transformed coordinates (x', y') of the original point (x, y) = (12, −4) would be calculated by: x' = x * scale factor y' = y * scale factor Calculating these: x' = 12 * \( \frac{1}{3} \) = 4 y' = −4 * \( \frac{1}{3} \) = −\( \frac{4}{3} \) Therefore, the image of the point (12, −4) after the dilation is (4, −\( \frac{4}{3} \)).
Dilation of a Point by Scale Factor at Origin
The question is asking to find the image of the point (12, -4) after a dilation by a scale factor of \( \frac{1}{3} \) centered at the origin. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The scale factor determines how much larger or smaller the image will be compared to the original figure. When the center of dilation is the origin (0,0), the image of a point can be found by multiplying each coordinate of the point by the scale factor. Given the point (12, -4) and the scale factor of \( \frac{1}{3} \), you multiply each coordinate of the point by \( \frac{1}{3} \): \( x' = x \cdot \text{scale factor} = 12 \cdot \frac{1}{3} = 4 \) \( y' = y \cdot \text{scale factor} = -4 \cdot \frac{1}{3} = -\frac{4}{3} \) Therefore, the image of the point (12, -4) after the dilation is (4, -\(\frac{4}{3}\)).
Dilation of a Point
To perform a dilation on a point with a scale factor centered at the origin, you simply multiply the x and y coordinates of the point by the scale factor. For the point (6, 12) and a scale factor of \( \frac{1}{2} \), the calculation would be as follows: - The x-coordinate after dilation: \( 6 \times \frac{1}{2} = 3 \) - The y-coordinate after dilation: \( 12 \times \frac{1}{2} = 6 \) Therefore, the image of the point (6, 12) after a dilation with a scale factor of \( \frac{1}{2} \) centered at the origin is (3, 6).
Dilation of a Point with Scale Factor and Origin
To find the image of a point after a dilation with a scale factor, you multiply each coordinate of the original point by the scale factor. When the dilation is centered at the origin (0, 0), this operation is straightforward. The original point is (-6, 7), and the scale factor is 5. Therefore, you multiply each coordinate of the point by the scale factor: New x-coordinate: -6 * 5 = -30 New y-coordinate: 7 * 5 = 35 So the image of the point (-6, 7) after dilation by a scale factor of 5, centered at the origin, is (-30, 35).
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