Question

problem 3
quadrilateral abcd is dilated with the origin as the center of dilation, moving point a onto point b.
heres your sketch from the previous screen.
write the coordinates for a, c, and d.

Explanation:

Step1: Recall dilation formula

For a dilation with center at the origin $(0,0)$ and scale - factor $k$, the transformation of a point $(x,y)$ to $(x',y')$ is given by $(x',y')=(kx,ky)$. First, we need to find the scale - factor $k$. The distance from the origin to point $A(2,2)$ and the distance from the origin to point $A'(4,4)$. The distance formula from the origin $(0,0)$ to a point $(x,y)$ is $d = \sqrt{x^{2}+y^{2}}$. The distance from the origin to $A(2,2)$ is $d_{A}=\sqrt{2^{2}+2^{2}}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$, and the distance from the origin to $A'(4,4)$ is $d_{A'}=\sqrt{4^{2}+4^{2}}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}$. The scale - factor $k=\frac{d_{A'}}{d_{A}} = 2$.

Step2: Dilate point $C$

Given point $C(-2,2)$. Using the dilation formula $(x',y')=(kx,ky)$ with $k = 2$, we have $x'=2\times(-2)=-4$ and $y'=2\times2 = 4$. So, $C'(-4,4)$.

Step3: Dilate point $D$

Given point $D(-2,-2)$. Using the dilation formula $(x',y')=(kx,ky)$ with $k = 2$, we have $x'=2\times(-2)=-4$ and $y'=2\times(-2)=-4$. So, $D'(-4,-4)$.

Answer:

$A'(4,4)$, $C'(-4,4)$, $D'(-4,-4)$