Question

a square with vertices at (1,1), (1,3), (3,3), and (3,1) is dilated with a scale factor of 2 using the origin as the center. what are the coordinates of the new vertices?
a. (2,2),(2,4),(4,4),(4,2)
b. (2,2),(2,6),(6,6),(6,2)
c. (0.5,0.5),(0.5,1.5),(1.5,1.5),(1.5,0.5)
d. (1,2),(1,4),(3,4),(3,2)

Explanation:

Step1: Recall dilation formula

When dilating a point $(x,y)$ with a scale - factor $k$ centered at the origin, the new coordinates $(x',y')$ are given by $(x',y')=(k\cdot x,k\cdot y)$. Here $k = 2$.

Step2: Dilate the first vertex $(1,1)$

For the point $(1,1)$, $x = 1,y = 1,k = 2$. Then $x'=2\times1 = 2,y'=2\times1 = 2$. The new point is $(2,2)$.

Step3: Dilate the second vertex $(1,3)$

For the point $(1,3)$, $x = 1,y = 3,k = 2$. Then $x'=2\times1 = 2,y'=2\times3 = 6$. The new point is $(2,6)$.

Step4: Dilate the third vertex $(3,3)$

For the point $(3,3)$, $x = 3,y = 3,k = 2$. Then $x'=2\times3 = 6,y'=2\times3 = 6$. The new point is $(6,6)$.

Step5: Dilate the fourth vertex $(3,1)$

For the point $(3,1)$, $x = 3,y = 1,k = 2$. Then $x'=2\times3 = 6,y'=2\times1 = 2$. The new point is $(6,2)$.

Answer:

B. $(2,2),(2,6),(6,6),(6,2)$