Question

consider gk located at g (0,0) and k (4,4). describe the dilation of gk by a scale factor of 2 centered at (-2,-4)

Answer 1

Coordinates of g(0,0) and k(4,4).

If we have to dilate gk by a scale factor of 2 , it means we have to multiply each coordinate by 2.

g(0×2,0×2) =p(0,0)⇒ New coordinate

k(4×2,4×2)=q(8,8)⇒  New Coordinate

This is the procedure when we dilate a shape when center is Origin.

Now ,as you have given center is not origin.

It's center is (-2,-4).

So, we will use the following Formula.

If a point(x,y) and is dilated by a factor k and it's center is (m,n) then

New position=[m+k(x-m),n+k(y-n)]

So,applying the formula ,we get new coordinates of g,k

g(0,0)=g'[-2+2{0-(-2)},-4+2{0-(-4)}]=g'(2,4)

k(4,4)=k'[-2+2{4-(-2)},-4+2{4-(-4)}]=(10,12)

Answer 2

Solution: The dilation of gk by a scale factor of 2 centered at (-2,-4)  is $g'(2,4)$ and $k'(10,12)$.

Explanation:

It is given that the scale factor is 2 and center of dilation is (-2,-4).

If the center of dilation is (a,b) with scale factor k, then the dilation of a point P(x,y) is determined by the given formula.

$D(x,y)=(a+k(x-a),b+k(y-b))$     .....(1)

To find the g' substitute x=0, y=0, a=-2, b=-4 and k=2 in equation (1).

$g'=(-2+2(2),-4+2(4))\\g'=(2,4)$

To find the k' substitute x=4, y=4, a=-2, b=-4 and k=2 in equation (1).

$k'=(-2+2(4+2),-4+2(4+4))\\g'=(10,12)$

Therefore, the dilation of gk by a scale factor of 2 centered at (-2,-4) is $g'(2,4)$ and $k'(10,12)$.