Given:
The red figure dilated with a scale factor of
and the center of dilation is at the point (4,2) to get the green figure.
To find:
The coordinates of C' and A.
Solution:
If a figure is dilated with a scale factor k and the center of dilation is at the point (a,b), then
![(x,y)\to (k(x-a)+a,k(y-b)+b)](https://tex.z-dn.net/?f=%28x%2Cy%29%5Cto%20%28k%28x-a%29%2Ba%2Ck%28y-b%29%2Bb%29)
In given problem, the scale factor is
and the center of dilation is at (4,2).
...(i)
Let the vertices of red triangle are A(m,n), B(10,14) and C(-2,11).
Using (i), we get
![C(-2,11)\to C'(\dfrac{1}{3}(-2-4)+4,\dfrac{1}{3}(11-2)+2)](https://tex.z-dn.net/?f=C%28-2%2C11%29%5Cto%20C%27%28%5Cdfrac%7B1%7D%7B3%7D%28-2-4%29%2B4%2C%5Cdfrac%7B1%7D%7B3%7D%2811-2%29%2B2%29)
![C(-2,11)\to C'(\dfrac{1}{3}(-6)+4,\dfrac{1}{3}(9)+2)](https://tex.z-dn.net/?f=C%28-2%2C11%29%5Cto%20C%27%28%5Cdfrac%7B1%7D%7B3%7D%28-6%29%2B4%2C%5Cdfrac%7B1%7D%7B3%7D%289%29%2B2%29)
![C(-2,11)\to C'(-2+4,3+2)](https://tex.z-dn.net/?f=C%28-2%2C11%29%5Cto%20C%27%28-2%2B4%2C3%2B2%29)
![C(-2,11)\to C'(2,5)](https://tex.z-dn.net/?f=C%28-2%2C11%29%5Cto%20C%27%282%2C5%29)
Therefore, the coordinates of Point C' are C'(2,5).
We assumed that point A is A(m,n).
Using (i), we get
![A(m,n)\to A'(\dfrac{1}{3}(m-4)+4,\dfrac{1}{3}(n-2)+2)](https://tex.z-dn.net/?f=A%28m%2Cn%29%5Cto%20A%27%28%5Cdfrac%7B1%7D%7B3%7D%28m-4%29%2B4%2C%5Cdfrac%7B1%7D%7B3%7D%28n-2%29%2B2%29)
From the given figure it is clear that the image of point A is (8,4).
![A'(\dfrac{1}{3}(m-4)+4,\dfrac{1}{3}(n-2)+2)=A'(8,4)](https://tex.z-dn.net/?f=A%27%28%5Cdfrac%7B1%7D%7B3%7D%28m-4%29%2B4%2C%5Cdfrac%7B1%7D%7B3%7D%28n-2%29%2B2%29%3DA%27%288%2C4%29)
On comparing both sides, we get
![\dfrac{1}{3}(m-4)+4=8](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B3%7D%28m-4%29%2B4%3D8)
![\dfrac{1}{3}(m-4)=8-4](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B3%7D%28m-4%29%3D8-4)
![(m-4)=3(4)](https://tex.z-dn.net/?f=%28m-4%29%3D3%284%29)
![m=12+4](https://tex.z-dn.net/?f=m%3D12%2B4)
![m=16](https://tex.z-dn.net/?f=m%3D16)
And,
![\dfrac{1}{3}(n-2)+2=4](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B3%7D%28n-2%29%2B2%3D4)
![\dfrac{1}{3}(n-2)=4-2](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B3%7D%28n-2%29%3D4-2)
![(n-2)=3(2)](https://tex.z-dn.net/?f=%28n-2%29%3D3%282%29)
![n=6+2](https://tex.z-dn.net/?f=n%3D6%2B2)
![n=8](https://tex.z-dn.net/?f=n%3D8)
Therefore, the coordinates of point A are (16,8).