Question

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

Answer 1

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

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: $(x,y)\Rightarrow (kx,ky),$ where $k$ is the scale factor.
Given scale factor $=0.5$
Coordinates of the square after dilation $=A^{\prime }(1\times 0.5,1\times 0.5),B^{\prime }(3\times 0.5,1\times 0.5),C^{\prime }(3\times 0.5,3\times 0.5),D^{\prime }(1\times 0.5,3\times 0.5)$
$=A^{\prime }(0.5,0.5),B^{\prime }(1.5,0.5),C^{\prime }(1.5,1.5),D^{\prime }(0.5,1.5)$Hence, the correct answer is option B.