How to find the matrix associated with a dilation operation? Recall that a point (x,y) is mapped by a dilation with a factor k=0 and center to .
O
(kx,ky)
Suppose the matrix we are looking for is (rtsu).
Find r,s,t,u such that it satisfies
(rtsu)(xy)=(kxky)
From the matrix multiplication on the left side, we get:
(rx+sytx+uy)=(kxky)
By equating the corresponding components:
First row:rx+sy=kx. For this equation to hold for all x and y, the coefficients of x must be equal, and the coefficients of y must be equal. Thus, r=k and s=0.
Determine the image of point A(2,4) transformed by a dilation with a factor of 2 with respect to the center point P(1,1)!
Alternative Solution:
Given x=2,y=4,k=2,a=1,b=1.
(x′y′)=(2002)(2−14−1)+(11)
(x′y′)=(2002)(13)+(11)
(x′y′)=((2)(1)+(0)(3)(0)(1)+(2)(3))+(11)
(x′y′)=(26)+(11)=(37)
Thus, the image of point A(2,4) is A′(3,7).
Visualization of Dilation of Point A(2,4) with Center P(1,1) and Scale Factor k=2
Point A(2,4) is dilated with respect to the center P(1,1) with a scale factor k=2 to produce the image A′(3,7). The line from the center to the original point and from the center to the image lie on the same line, and the distance PA′ is twice the distance PA.
Find the coordinates of the image of the point (a,b) under the dilation [O,3]!
Determine the matrix corresponding to a dilation with a scale factor of −2 and centered at O(0,0).
A point B(−1,5) is dilated with center P(2,−3) and scale factor k=21. Determine the coordinates of the image of point !
A triangle KLM with vertices K(1,1), L(5,1), and M(3,4) is dilated with center and scale factor . Draw the original triangle and its image, then determine the coordinates of the image vertices!
