# Points A and B are at #(4 ,1 )# and #(8 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # and dilated about point C by a factor of #1/2 #. If point A is now at point B, what are the coordinates of point C?

##### 2 Answers

The point

#### Explanation:

The points are the following

After the rotation counterclockwise about the origin, we get the point

Let the point

Then,

Therefore,

and

The point

graph{((x-4)^2+(y-1)^2-0.01)((x-8)^2+(y-3)^2-0.01)((x+1)^2+(y-4)^2-0.01)((x+10)^2+(y-5)^2-0.01)=0 [-14, 18.06, -5.07, 10.93]}

#### Explanation:

Under a counterclockwise rotation about the origin of

#pi/2#

#• " a point " (x,y)to(-y,x)#

#rArrA(4,1)toA'(-1,4)" where A' is the image of A"#

#"Under a dilatation about C of factor "1/2# Taking a

#color(blue)"vector approach"#

#rArrvec(CB)=1/2vec(CA')#

#rArrulb-ulc=1/2(ula'-ulc)#

#rArrulb-ulc=1/2ula'-1/2ulc#

#rArr1/2ulc=ulb-1/2ula#

#color(white)(rArr1/2c)=((8),(3))-1/2((-1),(4))#

#color(white)(raArr1/2c)=((17/2),(1))#

#rArrulc=2((17/2),(1))=((17),(2))#

#rArrC=(17,2)#