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0
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RainbowPanda  Apr 14, 2018
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#1
+85726
+2

Here is a formula that will always work for these kind of problems

New Point  =  [ Old Point - Dilation Center ] * Scale Factor  + Dilation Center

Note that  (a, b) + (c - d)  = (a + c, b + d)

And that (a,b) - (c,b)  = (a - c, b - d)

Also  (a,b)*constant  =  (a*constant , b*constant )

So we have

New Point   =  [ (4,7) - (1,2) ] *10 + (1,2)  =

[(3, 5)]*10  + (1.2)  =

(30, 50)  + (1,2) =

(31, 52)

CPhill  Apr 15, 2018
#2
+1862
+2

In order to dilate a point about a center of dilation that is not the origin, I would recommend manipulating the current Cartesian plane where the origin lies at the center of dilation.

For example, if we translate the entire coordinate plane by the rule \((x,y)\Rightarrow (x-1,y-2)\), then the center of dilation becomes the origin, and the point (4,7), after the translation has occurred, transforms to (3,5).

We can now dilate the point like normal because the origin is now the center of dilation. I can now use the scale factor where the dilated point can be determined as \((x,y)\Rightarrow (kx,ky)\) , where k, 10 in this case, is the scale factor.

Therefore,\((3,5)\Rightarrow(10*3,10*5)=(30,50)\).

We aren't done yet, though! Remember the translation we did at the beginning? We have to undo this action to determine the true location on the real Cartesian plane. Let's translate \((x,y)\Rightarrow(x+1,y+2)\) to undo the previous translation. Therefore, \((30,50)\Rightarrow (31,52)\) , or the second option listed.

TheXSquaredFactor  Apr 15, 2018

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