A dilation has center $(0,0)$. Find the image of each point for the given scale factor. $A(3,4) ; D_7(A)$
A. $(10,11)$
B. $(28,21)$
C. $(21,28)$
D. $(11,10)$
Answer (2)
Understand that dilation scales the coordinates of a point by the scale factor.
Multiply the x-coordinate of the original point by the scale factor: 3 × 7 = 21 .
Multiply the y-coordinate of the original point by the scale factor: 4 × 7 = 28 .
The image of the point is ( 21 , 28 ) .
Explanation
Understanding the Dilation We are given a point A ( 3 , 4 ) and asked to find its image after a dilation with center ( 0 , 0 ) and scale factor 7 . A dilation scales the coordinates of a point by the scale factor.
Calculating the Image Coordinates To find the image of point A after the dilation, we multiply both the x and y coordinates of A by the scale factor 7 .
The x -coordinate of the image is 3 × 7 = 21 .
The y -coordinate of the image is 4 × 7 = 28 .
Finding the Image Therefore, the image of point A ( 3 , 4 ) after the dilation D 7 is ( 21 , 28 ) .
Examples
Dilations are used in creating scaled maps and architectural designs. For example, if an architect wants to create a blueprint of a building that is 100 times smaller than the actual building, they use dilation with a scale factor of 1/100. Similarly, when creating a map, the distances on the map are scaled down versions of the actual distances on the ground.
The image of the point A(3,4) after dilation with a scale factor of 7 is (21, 28). The calculations involve multiplying the x and y coordinates of A by the scale factor. The correct multiple-choice option is (C) (21, 28).
;
