Question

Square $ABCD$ is dilated about the origin. If the scale factor is $0.5$, what are the coordinates of the resultant image $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ ?

• A. $A^{\prime }(0.5,0.5),B^{\prime }(1.5,0.5),C^{\prime }(1.5,1.5),D^{\prime }(0.5,1.5)$
• B. $A^{\prime }(2,2),B^{\prime }(6,2),C^{\prime }(6,6),D^{\prime }(2,6)$
• C. $A^{\prime }(0.25,0.25),B^{\prime }(1.5,0.25),C^{\prime }(1.5,1.5),D^{\prime }(0.25,1.5)$
• D. $A^{\prime }(3,3),B^{\prime }(1.5,3),C^{\prime }(1.5,1.5),D^{\prime }(3,1.5)$

Answer 1

The correct option is A $A^{\prime }(0.5,0.5),B^{\prime }(1.5,0.5),C^{\prime }(1.5,1.5),D^{\prime }(0.5,1.5)$

Coordinates of the square $=A(1,1),B(3,1),C(3,3),D(1,3)$
The size by which the shape is enlarged or reduced is called its scale factor.
If the center of dilation is the origin, then the coordinates are multiplied by the scale factor to get the coordinates of the dilated image:
$(x,y)\Rightarrow (kx,ky)$, where k is the scale factor.
Given, scale factor $=0.5$
Coordinates of the square after dilation = $A^{\prime }(1\times 0.5,1\times 0.5),B^{\prime }(3\times 0.5,1\times 0.5),C^{\prime }(3\times 0.5,3\times 0.5),D^{\prime }(1\times 0.5,3\times 0.5)$
$=A^{\prime }(0.5,0.5),B^{\prime }(1.5,0.5),C^{\prime }(1.5,1.5),D^{\prime }(0.5,1.5)$