A dilation has center $(0,0)$. Find the image of each point for the given scale factor. $P(-3,5) ; D_1(P)$
A. $(5,-3)$
B. $(-2,6)$
C. $(6,-2)$
D. $(-3,5)$
Answer (1)
A dilation centered at the origin with a scale factor k transforms a point ( x , y ) to ( k x , k y ) .
With a scale factor of 1, the transformation becomes D 1 ( x , y ) = ( 1 ⋅ x , 1 ⋅ y ) = ( x , y ) .
Applying this to the point P ( − 3 , 5 ) , we have D 1 ( − 3 , 5 ) = ( − 3 , 5 ) .
Therefore, the image of P ( − 3 , 5 ) after the dilation is ( − 3 , 5 ) .
Explanation
Analyze the problem We are given a point P ( − 3 , 5 ) and asked to find its image after a dilation centered at the origin ( 0 , 0 ) with a scale factor of 1 .
Apply the dilation transformation A dilation with center ( 0 , 0 ) and scale factor k transforms a point ( x , y ) to ( k x , k y ) . In this case, the scale factor is k = 1 . Therefore, the transformation is: D 1 ( x , y ) = ( 1 ⋅ x , 1 ⋅ y ) = ( x , y ) This means that the image of the point is the same as the original point.
Calculate the image of the point Applying the dilation to the point P ( − 3 , 5 ) , we get: D 1 ( − 3 , 5 ) = ( 1 ⋅ ( − 3 ) , 1 ⋅ 5 ) = ( − 3 , 5 ) So, the image of P ( − 3 , 5 ) after the dilation is ( − 3 , 5 ) .
Select the correct answer Comparing the result ( − 3 , 5 ) with the given options, we see that option D is the correct answer.
Examples
Dilations are used in various real-world applications, such as scaling images in graphic design, creating architectural models, and understanding how objects appear at different distances. For example, when you zoom in on a digital map, you are essentially applying a dilation to enlarge the map around a central point. Similarly, artists use dilations to create perspective in their drawings, making objects appear larger or smaller depending on their distance from the viewer. Understanding dilations helps in fields like photography, where adjusting the focal length of a lens changes the size and scale of the objects in the image.
